Apeirostikc LogismcMia Pragmatik Metablht

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2

Prokatarktik1. , , . . ( ) , . - - - . . . 2. . , , . , . , Bolzano - Weierstrass . , . 3. ( , ) . , - : , . 4. - - : ( ), , . . . , , 3

, . . , y = xn , x log x = 1 1 dt t . , , . 5. - n0 - - . n0 . . 6. Riemann Darboux. . Darboux Riemann . 7. . , . - ! 8. 2007 - 2008 . , , , , , . , , . 13 2008.

4

Perieqmena1 1.1 . . . . . . . . . . . . . . . 1.2 . . . . . . . . . . . . . . . . 1.3 . . . . . . . . . . . . . . 1.4 . . . . . . . . . . . . . . . . . . . . 1.5 . 2 2.1 . . . . . . . . . . . . . . . . . . 2.2 . . . . . . . . . . . . . 2.3 . . . . . . . 2.4 . 2.5 . e, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 16 24 31 32 41 41 46 50 54 70 79 79 82 86 87 100 105 108 110 112 114 120 125 125 139 143 158 159 162

3 3.1 . . . . . . . . . . . . . . . 3.2 . . . . . . . . . . . . . . . . . 3.3 . . . . . . . . . . . . . . . 3.4 . . . . . . . . . . . . . . . 3.5 . . . . . . . . . . . . . . . . . . . . . . . 3.6 . . . . . . . . . . . . . . . 3.7 . . . . . . . . . . . . . . . . . . . . . . 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 . . . . . . . . . . . . . . . 3.10 . 3.11 . . . . . 4 4.1 , . . . . . . . . 4.2 . . . . . . . . . . 4.3 . . . . . . . . . . . . 4.4 . 4.5 . . . . . . . . . . 4.6 . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7 4.8 4.9

. . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . . . . . 168 . . . . . . . . . . . . . . . . . . . . . . . . 171 177 177 182 184 186 192 198 199

5 5.1 , . . . . . . . . . 5.2 . . . 5.3 . 5.4 . . . . . . . 5.5 . 5.6 . . . . . . . . 5.7 . . . . . . .

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6 6.1 6.2 . . . . . . . . . . . . . . . . 6.3 , . . . . . . . 6.4 . . 6.5 . . . . . . . . 6.6 , . . . . . . 6.7 . . . . . 6.8 : . . 6.9 . . . 6.10 . . . . . . . . . . . 6.11 . 6.12 . . . . . . . 7 7.1 . . . . . . . . . . . . . . 7.2 . . . . . . . . . . 7.3 . . . . 7.4 .

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207 . 207 . 209 . 213 . 217 . 221 . 231 . 233 . 239 . 245 . 257 . 258 . 266 269 269 278 283 293 317 317 324 331 349

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8 8.1 . . . 8.2 . . . . . . . . . . . 8.3 . . . . 8.4 . . . . . . . . . .

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9 353 9.1 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.2 . . . . . . . . . . . . . . . . . . 357 9.3 . . . . . . . . . . . . . . . . . . . . . . 359 6

10 10.1 . . 10.2 . 10.3 . . . 10.4 . . . . . . . . . . . 10.5 Taylor. . . . . . . . . . .

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365 365 369 374 379 387

7

8

Keflaio 1

Oi Pragmatiko Arijmo - - . , , . , , . , , . , .

1.1 H pragmatik eujea. . , , . . , , . x + y, x y, xy x (y = y 0) x, y . ( , .) . 1, 2, 3, . . . , 0, 1, 1, 2, 2, 3, 3, . . . , m , m, n n n = 0. m m , 1 . . , . , , , . . , 9

. , . : (1) : , m , 2m , 3m , . . . . n 2n 3n (2) m m n 4 n , . , 3 4 3 4 4 . 3 3 R . , N, Z Q , , . : 0, N 0, 1, 2, . . . N 1, 2, . . . . . , . . . 1.1 (1) x y y z, x z. , . (2) x y, x + z y + z x z y z. (3) x y z w, x + z y + w. , . (4) x y z > 0, xz yz x y . z z (5) x y z < 0, xz yz x y . z z (6) 0 < x y 0 < z w, 0 < xz yw. , . x |x| x, x 0, x, x 0. |x| = x, x 0, x, x 0.

, . . 1.2 (1) |xy| = |x||y|. (2) |x| |y| |x y| |x| + |y|. (3) y = 0, x = |x| . y |y| (4) |x| a a x a. (5) |x| < a a < x < a. x, y max{x, y} min{x, y}. : max{x1 , . . . , xn } min{x1 , . . . , xn }. A R , A 10

A, maximum A max A. , A , minimum A min A. . R. . 0. 1. . , x , x > 0, , x < 0, |x|. , : . , .

1.1: . x . : x x. , : . x 0 |x|. , , : x, y |x y|. , : x < y x, y 0, 1. , 1 0, 11

: x < y y x. 1 0, : x < y y x. : 1 0. , ( , ), 1 0. . . R. a < b, (a, b) = {x : a < x < b}, (a, b] = {x : a < x b} [a, b) = {x : a x < b}. a b, [a, b] = {x : a x b}. a, b. (a, b) [a, b] . (a, +) = {x : x > a}, (, b) = {x : x < b}, [a, +) = {x : x a} (, b] = {x : x b}. ( ) ( ) ( ). , (, +) = R, . : +, : . + () () . < , > , < x, x < +, < + x. , + () () ( ) . , . . .

1 b, , , . , , ( ). : b > 0 n n 1 > b . : 12

1.3 b > 0 n > b.

1.2: n > b. 1.3 . n b, n + 1, n + 2, n + 3, . . . b. . . : , .1 b = a , a , 1.3 :

1.4 . a > 0 n 1 n < a.

1.3:

1 n

< a.

1 , n 1 1 1 a, n+1 , n+2 , n+3 , . . . a. .

. : , . . 1.5 x k k x < k + 1. x [k, k + 1), k . . . . , [3, 2), [2, 1), [1, 0), [0, 1), [1, 2), [2, 3), . . . . 13

k k x < k + 1 x [x]. : [3] = 3, [4] = 4, [ 8 ] = 1, [ 2 ] = 0, [ 8 ] = 2. 5 3 5 . .

. . 1.6 Q R. a, b r a < r < b. a < b

. . , a, b a < b x a < x < b, : , x , x = a+b . , a+b 2 2 a b. , a, b a+b . 2 a, b a+b . 2 Q R . . , x > 0, r x < r < x + r x < r < x.

Askseic., . 1. r a , r + a . 2. r = 0 a , ra . 3. a , p, q, r, s p + qa = r + sa, p = r q = s. , . 1. x y < 0 z w < 0, 0 < yw xz. 14

2. x y, z w, t s x + z + t = y + w + s, x = y, z = w t = s. 0 < x y, 0 < z w, 0 < t s xzt = yws, x = y, z = w t = s. 3. |x + y| = |x| + |y| x, y 0 x, y 0. |x + y + z| |x| + |y| + |z|. , |x + y + z| = |x| + |y| + |z| x, y, z 0 x, y, z 0. 4. t x t y t min{x, y}. t x t y t max{x, y}. 5. max{x, y} =x+y+|xy| 2 x+y|xy| 2

min{x, y} =

.

6. ; [a, b], (a, b), [a, b), N, Z, Q, 1 : n . n

. 1. ; 2. . . a x b a y b, |x y| b a. a < x < b a < y < b, |x y| < b a. 3. , x, y, x + y, x y, xy x . y ( xy : x, y > 0. 0 . , 1, y, . , . , ;) . 1. x . |x + 1| > 2, |x 1| < |x + 1|, x x+3 > , x+2 3x + 1 (x 2)2 4,

|x2 7x| > x2 7x,

(x 1)(x + 4) > 0, (x 7)(x + 5) 15

(x 1)(x 3) 0. (x 2)2

2. x x . (, 3], (2, +), (3, 7), (, 2) (1, 4) (7, +), (, 2] [1, 4) [7, +).

[2, 4] [6, +), .

[1, 4) (4, 8],

1. x [x] = [x] ; 2. k , [x + k] = [x] + k. 3. [x+y] = [x]+[y] [x+y] = [x]+[y]+1 . [x + y + z]. 4. , 0 < x 1, n 1 1 n+1 < x n n x. 5. l > 0. n > l. n l; l n n l; a > 0. n 1 n < a. n a; . 1. x < x > x . 2. .

1.2 Dunmeic kai rzec.. . an ( ) n an = a a ,n

16

n a. a = 0, a0 an n a0 = 1, an = 1 1 = . an aan

(a)n = an , n , (a)n = an , n . , , , n , an > 0 a = 0 , n , (i) an > 0 a > 0 (ii) an < 0 a < 0. . 1.7 n 2, xn y n = (x y)(xn1 + xn2 y + + xy n2 + y n1 ). , n 3, xn + y n = (x + y)(xn1 xn2 y + xy n2 + y n1 ). n 1 2 (n 1) n n! . n! = 1 2 (n 1) n. : 1! = 1, 2! = 1 2 = 2, 3! = 1 2 3 = 6, 4! = 1 2 3 4 = 24. , 0! = 1 n n! = (n 1)! n. m, n 0 m n n n! = . m m!(n m)! :n 0 n m

-

=

n n

= 1,

n 1

=

n n1

= n,

n 2

=

n n2

=

n(n1) 2

.

1 m n, , , n n(n 1) (n m + 1) = . m m! 17

1.8 Newton. x, y n (x + y)n = n n n n1 x + x y + + 0 1 n n1 xy n1 + n n y . n

: (x + y)1 = x + y , (x + y)2 = x2 + 2xy + y 2 , (x + y)3 = x3 + 3x2 y + 3xy 2 + y 3 , (x + y)4 = x4 + 4x3 y + 6x2 y 2 + 4xy 3 + y 4 , (x + y)5 = x5 + 5x4 y + 10x3 y 2 + 10x2 y 3 + 5xy 4 + y 5 Newton. . . , . . 1.1 n a > 0, xn = a x > 0.

1.1 xn = a a > 0 . , , . 1.9 (1) n , xn = a (i) , , a > 0, (ii) , 0, a = 0, (iii) , a < 0. (2) n , xn = a (i) , , a > 0, (ii) , 0, a = 0, (iii) , , a < 0. n , a xn = a n- a n a. n , a 0 xn = n- a a n a . , , n 0 = 0 n a > 0 a > 0 n. n , a < 0 n a < 0 n n a n. 18

n = 2, 3, 4, . . . , n a , , , . . . a. n = 2 2 a a , , a. n = 3 3 a a. : (1) x4 = 16 , 4 16 = 2 4 16 = 2. , x4 = 16 . (2) x5 = 32 , 5 32 = 2. x5 = 32 , 5 32 = 2. 5 32 2 = 5 32. 5 32 = x5 = 32, 5 32 x5 = 32. , , : n a = n a a n. . 1.10 n, k. n k k n- . : (1) 2 , , m m2 = 2. , 3 5 m m3 = 5. (2) 2 + 3 . , r, 2 5 ( 2 + 3)2 = r2 , 6 = r 2 , , 6 . m m2 = 6. . .

ar r . r r = m , m , n n , , m, n , 1. r . : 16 8 10 5 . , 6 3 . 4 2 ar = ( n a)m

0, : (i) a > n a , (ii) a = 0, n 0 = 0, m > 0 , = , r > 0 0r = ( n 0)m 0m = 0 (iii) a < 0, n n a . : ar (i) a > 0, (ii) a = 0 r > 0 (iii) a < 0 r . 19

3 6 3 3 : (1) 2 4 = ( 4 2)3 , 2 8 = 2 4 = ( 4 2)3 , 2 4 = ( 4 2)3 =6 2

(

1 4 2)3

,

2 = 2 = 8 2 =2 = = . 3 3 3 3 4 = 0 0 5 = 0. 0 4 , 0 5 00 . (2) 0 5 10 5 5 (3) (2) 3 = ( 3 2)5 = ( 3 2) = ( 3 2)5 , (2) 6 = (2) 3 (2)0 = 1. 5 14 (2) 2 (2) 4 . : (1) n 1 1 1 1 n . n n . a n n a : an =1

3

6 2

3

1 23

1 8

n

a.

(2) , ar a < 0 : r. , , , . ar a < 0 r . ar , ar > 0 r a > 0. : , . 1.11 . . .

, ax a 0 x . a > 1. , r, s, t s < r < t, , , as < ar < at . , , , s, t x s < x < t, as < ax < at . , as , at ax . , ax : as < ax < at s, t s < x < t. . s < x t > x. s, t , , s < t a > 1, as < at . , , , as , , at , 20

( ). . , , as at : as < < at s, t s < x < t. , = as < < at s, t s < x < t. , , . 1.11 a > 1 x. as < < at s, t s < x < t. a > 1 x , ax 1.11. , , ax as < a x < a t s, t s < x < t . a = 1 x , 1x = 1. , 0 < a < 1 x , 1 . ( a )x : ax = 1 ax 1 a

> 1 x

.

, x , 0x = 0. : (1) , x , ax , (i) a > 0 (ii) a = 0 x > 0. ax , (i) a < 0 x (ii) a = 0 x < 0. (2) , ax , (i) a > 0 x , (ii) a = 0 x > 0 (iii) a < 0 x . ax : x a > 0 ( a < 0), ( ax ) ax > as s < x, , as > 0, ax > 0. . 21

1.12 (1) a > 0, ax ay = ax+y , (ax )y = (ay )x = axy , ax bx = (ab)x .

(2) 0 < a < b, (i) ax < bx , x > 0, (ii) a0 = b0 = 1 (iii) ax > bx , x < 0. (3) x < y, (i) ax < ay , a > 1, (ii) 1x = 1y = 1 (iii) ax > ay , 0 < a < 1.

Askseic. . 1. n . n , xn < y n x < y. n , xn < y n |x| < |y|. 2. x, y = 0, x2 + xy + y 2 > 0 x4 +x3 y+x2 y 2 +xy 3 +y 4 > 0. ; x3 + x2 y + xy 2 + y 3 > 0 x5 + x4 y + x3 y 2 + x2 y 3 + xy 4 + y 5 > 0; ; 3. n (i) 1 + 2 + + n =2 2

(ii) 1 + 2 + + n = (iii) 1 + 2 + +3 3

1 6 n(n + 1)(2n + n3 = 1 n2 (n + 1)2 . 4

1 2 2

n(n + 1), 1),

4. Newton Pascal; 1 1 1 1 1 1 4 2 1 3 3 6 4 1 1

n n 1 m n, n+1 = m + m1 . m Pascal;

Newton n, ; 22

5.

n m n m

, n m ; , m 0 n;

Pascal; . 1. n , n an = a. n , n an = |a|. 2. a + b a+ b a, b 0. a + b = a + b a = 0 b = 0. 3. : n n n a b = ab ,n m

a=

m

n

a=

nm

a.

; , a < 0 b < 0 ; 4. n 0 a < b, n a < n b . n a < b, n a < n b . 5. 105 105 106 106 . 6. 7 129 , 3 5 + 2 3 2 + 5 . 7. R, a, b a < b x a < x < b. (: a + 2 , b + 2 .) 8. a > 0, a a a a=2n

a a

,

n . 9. a. (: 1 < a. , 0, 1 a, . x , x2 = a.) 4 a 8 a . .7 16 10 4 14

1. (2)0 , 00 , (3) 3 , (2) 12 , (2) 12 ; (8) 3 , (1) 6 . 2. (1) 32 5 4

= (1) 3 4 ; 23

2 5

3. r (2)r < 0; . 2 2

1. 2

, (2)

, 0

2. ( 17 3)24 < 3 2

,0 2 ; < ( 17 3)25 .2 5

2

3. 2

3

3

. 2

4. [10 2 5. (1)2 3

2

] [100 2 3

].

= (1)2

;

1.3 Dekadik anaptgmata.. . x = 48305. ( ) 48305 4 10000 + 8 1000 + 3 100 + 0 10 + 5 = 4 104 + 8 103 + 3 102 + 0 10 + 5. , , 10 0, 1, . . . , 9. . 4, 8, 3, 0, 5 x = 48305 48305 x. , 10 , . 1.13 x x = XN 10N + XN 1 10N 1 + + X1 10 + X0 XN , . . . , X0 {0, 1, . . . , 9} XN = 0. XN , . . . , X0 x = XN 10N + XN 1 10N 1 + + X1 10 + X0 x XN XN 1 . . . X1 X0 x. . [0, 1). , 0, 25 2 5 25 1 2 101 + 5 102 = + = = . 10 100 100 4 24

, 0, 5403 5 101 + 4 102 + 0 103 + 3 104 = 5 4 0 3 5403 + + + = . 10 100 1000 10000 10000

0, 25 0, 5403 0, 25000 . . . 0, 5403000 . . . , 0 . 1 5403 4 10000 . 3 , , 7 0, 42857 . . . 0 . : 3 0, 42857 . . . ; , 7 , 4 101 + 2 102 + 8 103 + 5 104 + 7 105 + 3 . 7 . . . 3 0, 42857 . . . 7 3 7 . 0, 4 3 < 0, 5 , 7 0, 42 3 < 0, 43 , 7 0, 428 3 < 0, 429 , 7

0, 4285

3 3 < 0, 4286 , 0, 42857 < 0, 42858 , 7 7 ...............

0, 4, 0, 42, 0, 428, 0, 4285, 0, 42857, . . . 3 4, 2, 8, 5, 7, . . . 3 . 7 7 0, 4, 0, 42, 0, 428, 0, 4285, 0, 42857, . . . 3 ; , 7 3 1 1 1 1 1 7 10 , 102 , 103 , 104 , 105 , . . . 1 1 1 1 1, 2 , 3 , 4 , 5 , . . . . , 1 , n , . , 1 1 1 1 1 1 ( ) 10 , 102 , 103 , 104 , 105 , . . . , 10n , . 0, 4, 3 0, 42, 0, 428, 0, 4285, 0, 42857, . . . 7 . : . , 3 , 7 , 0, 25000 . . . 25

1 4

. , 1 < 0, 3 , 4 0, 25 1 < 0, 26 , 4 0, 250 1 < 0, 251 , 4

0, 2

0, 2500

1 1 < 0, 2501 , 0, 25000 < 0, 25001 , 4 4 ...............

, , 0, 35699999 . . . , 9 , 357 0, 35700000 . . . , 1000 . , 9 0 , , . . 1.14 x [0, 1) x1 , x2 , x3 , . . . , {0, 1, . . . , 9}, x1 x2 xn x1 x2 xn 1 + + + n x < + + + n + n 10 102 10 10 102 10 10 n. x1 , x2 , x3 , . . . x 9 . x x1 , x2 , . . . 1.14, x 0, x1 x2 x3 . . . , xn n- x sn = x1 x2 xn + 2 + + n 10 10 10

n- x. 1 sn x < sn + 10n , x n- , , , 0 x sn < 1 . 10n

1 10n . , , x sn . :

x sn . : , xsn 26

.

. x [0, 1). 1 x1 x < x1 + 10 x1 10x < x1 + 1 , 10 10 , x1 10x. x1 = [10x]. x x x x 1 1 x1 + 1022 x < x1 + 1022 + 102 , s1 + 1022 x < s1 + 1022 + 102 , 10 10 x2 102 (x s1 ) < x2 + 1 , , x2 102 (x s1 ). x2 = [102 (x s1 )]. , x1 , . . . , xn1 , (n 1)x sn1 = x1 + + 10n1 , n1 10 xn xn 1 sn1 + 10n x < sn1 + 10n + 10n xn 10n (x sn1 ) < xn + 1 , , xn = [10n (x sn1 )]. x : x1 x1 = [10x] n 2 xn x1 , . . . , xn1 ( x2 x1 , x3 x1 , x2 ) xn = [10n (x sn1 )]. x1 = [10x], xn = [10n (x sn1 )] (n 2),

n- . : x1 = 10 x2 = 102 x3 = 103 x4 = 104 x5 = 105 x6 = 106 13 16 13 16 13 16 13 16 13 16 13 16 = 13 16

.

65 8 4 = 8, s1 = = , 8 10 5 4 5 1 81 = = 1, s2 = s1 + 2 = , 5 4 10 100 81 5 2 203 = = 2, s3 = s2 + 3 = , 100 2 10 250 203 5 13 = [5] = 5, s4 = s3 + 4 = , 250 10 16 13 0 13 = [0] = 0, s5 = s4 + 5 = , 16 10 16 13 0 13 = [0] = 0, s6 = s5 + 6 = 16 10 16

. 13 0, 8125000 . . . . 16 s4 13 16 x5 , x6 , . . . 0. : 1 x1 = 10 = 7, 2 s1 = 7 , 101 2

.

27

1 x2 = 102 2 1 3 x3 = 10 2 1 x4 = 104 2 1 x5 = 105 2 1 6 x6 = 10 2

0 7 7 = 0, s2 = s1 + 2 = , 10 10 10 7 7 707 = 7, s3 = s2 + 3 = , 10 10 1000 1 7071 707 = 1, s4 = s3 + 4 = , 1000 10 10000 7071 7071 0 , = 0, s5 = s4 + 5 = 10000 10 10000 7071 6 707106 = 6, s6 = s5 + 6 = 10000 10 1000000

1 . 2 0, 707106 . . . .Ac dome thn apdeixh thc Prtashc 1.14. 'Estw 0 x < 1. Orzoume x1 = [10x], opte x1 10x < x1 + 1 kai, epomnwc,x1 1 x1 x< + . 10 10 10

Parathrome ti 0 10x < 10, opte o x1 ankei sto {0, 1, . . . , 9}. Katpin orzoume s1 = kai x2 = [102 (x s1 )]. 'Ara x2 102 (x s1 ) < x2 + 1, optex1 x2 x2 x1 1 + 2 x< + 2 + 2. 10 10 10 10 10

x1 10

Epeid 0 102 (x s1 ) < 10, o x2 ankei sto {0, 1, . . . , 9}. Katpin orzoume s2 = kai x3 = [103 (x s2 )]. 'Ara x3 103 (x s2 ) < x3 + 1, optex2 x2 x1 x3 x1 x3 1 + 2 + 3 x< + 2 + 3 + 3. 10 10 10 10 10 10 10

x1 10

+

x2 102

Epeid 0 103 (x s2 ) < 10, o x3 ankei sto {0, 1, . . . , 9}. Aut h aperiristh epagwgik diadikasa dhmiourge touc xn , ton na met ton llo, gia kje n. Dhlad stw ti qoume bre x1 , . . . , xn1 ap to {0, 1, . . . , 9} tsi ste na enaix1 xn1 xn1 x1 1 + + n1 x < + + n1 + n1 . 10 10 10 10 10

Tte orzoume sn1 =10n (x

sn1 ) < xn + 1, opte

x1 10

+ +

xn1 10n1

kai xn = [10n (x sn1 )]. Aut shmanei ti xn

x1 xn1 xn1 xn x1 xn 1 + + n1 + n x < + + n1 + n + n . 10 10 10 10 10 10 10

Epeid 0 10n (x sn1 ) < 10, o xn ankei sto {0, 1, . . . , 9}. 'Estw ti uprqoun kai oi arijmo y1 , y2 , . . . , loi sto {0, 1, . . . , 9}, ste na isqeiy1 yn1 yn y1 yn1 yn 1 + + n1 + n x < + + n1 + n + n 10 10 10 10 10 10 10

gia kje n. Ap tic duo teleutaec diplc anisthtec enai 1 < (y1 10n1 + + yn1 10 + yn ) (x1 10n1 + + xn1 10 + xn ) < 1 kai, epeid o (y1 10n1 + + yn1 10 + yn ) (x1 10n1 + + xn1 10 + xn ) enai akraioc, sunepgetaiy1 10n1 + + yn1 10 + yn = x1 10n1 + + xn1 10 + xn

gia kje n. Efarmzontac aut thn isthta tan o n parnei tic timc 1, 2, 3, . . . , brskoume diadoqik y1 = x1 , y2 = x2 , y3 = x3 , . . . .

28

Ac upojsoume, tloc, ti ap kpoio shmeo kai pra loi oi xn enai soi me 9. Dhlad ti uprqei kpoioc fusikc m ste na enai xn = 9 gia kje n m. Ttex1 xm1 9 9 x1 xm1 9 9 1 + + m1 + m + + n x < + + m1 + m + + n + n 10 10 10 10 10 10 10 10 109 gia kje n m. Epeid 10m + + 1 1 10n , sunepgetai 10m1 9 10n

=

9 (10nm 10n

+ + 10 + 1) =

10nm+1 1 10n

=

xm1 1 1 x1 xm1 1 x1 + + m1 + m1 n x < + + m1 + m1 10 10 10 10 10 10 101 1 gia kje n m. Orzoume a = x1 + + 10m1 + 10m1 x, opte enai 0 < a 10n gia m1 10 1 kje n m. Epeid oi 10n gnontai mikrteroi ap kje jetik arijm, sunepgetai ti ap 1 1 kpoion n kai pra ja enai 10n < a. Aut enai topo afo enai a 10n gia kje n m. x

.

.

[0, 1). x 1, x = [x]+(x[x]), [x] x, , x [x] [0, 1). : XN . . . X0 [x] 0, x1 x2 . . . x [x]. , XN . . . X0 , x1 x2 . . . x. n- x sn = [x] + x1 xn x1 xn + + n = XN 10N + + X1 10 + X0 + + + n . 10 10 10 10 1 10n

, sn x < s n +

1 n. 0 x sn < 10n , , x [0, 1) x 0.

x sn . : , xsn . tn = sn + , , sn x < t n 29 tn sn = 1 . 10n 1 , 10n

sn+1 = sn + tn+1 = = sn+1 + sn + tn . 1 10n+1 9 + = sn + 1 10n+1 xn+1 1 + n+1 n+1 10 10 1 = sn + n 10 xn+1 sn 10n+1

10n+1

, , s1 , s2 , s3 , . . . ( x) t1 , t2 , t3 , . . . ( 1 > x). tn sn = 10n tn sn . x sn , tn , x sn ( ) tn x , , . , , . sn x sn tn tn x .

Askseic. . 1. x N . XN , . . . , X0 {0, 1, . . . , 9} XN = 0, 10N XN 10N + + X1 10 + X0 10N +1 1 < 10N +1 . x N + 1 10N x < 10N +1 . [0, 1). 1. 3 7

. ;

2. x, y [0, 1). n n- 1 x, y , |x y| < 10n . 3. x [0, 1). x1 , x2 , . . . x sn n- , 10n (x sn ); 4. x, y [0, 1). x + y x, y n- 2 10n . xy 2 10n 101 . 2n 30

. 1. 2 .

1.4 Logrijmoi. a (0, 1) (1, +), = 1. : y ax = y ( x) ; x ax > 0, , x a = y, y > 0. y. 1.2 a > 0 a = 1. y > 0 ax = y. x

1.2 ax = y. . , x1 , x2 ax = y ( y), , x1 = x2 , ax1 = ax2 . ax = y y a loga y. , : x = loga y ax = y.

: a = 1, ax = y, . , 1x = 1 x, y 1 , . a = 0 . 0x = y y = 0 , . a < 0 ax x ax x . . 1.15 a > 0 a = 1. (1) loga (yz) = loga y + loga z y, z > 0. (2) loga y = loga y loga z y, z > 0. z (3) loga (y z ) = z loga y y > 0 z. (4) loga 1 = 0 loga a = 1. (5) 0 < y < z. (i) loga y < loga z, a > 1, (ii) loga y > loga z, 0 < a < 1. 31

(1) Orzoume x = loga y kai w = loga z , opte ax = y kai aw = z . Tte ax+w = ax aw = yz , opte loga (yz) = x + w = loga y + loga z . (2) Ap thn loga y + loga z = loga ( y z) = loga y sunepgetai loga y = loga y loga z . z z z (3) Orzoume x = loga y , opte ax = y . Tte azx = (ax )z = y z kai, epomnwc, loga (y z ) = zx = z loga y . (4) H loga 1 = 0 prokptei ap thn a0 = 1 kai h loga a = 1 ap thn a1 = a. (5) 'Estw 0 < y < z . Orzoume x = loga y kai w = loga z , opte y = ax kai z = aw . Tte ax < aw kai, an a > 1, sunepgetai x < w en, an 0 < a < 1, sunepgetai x > w.

. 1.16 a, b > 0 a, b = 1. logb y = y > 0.'Estw a, b > 0 kai a, b = 1. Orzoume x = logb y kai w = loga b, opte bx = y kai aw = b. Sunepgetai awx = (aw )x = bx = y . 'Ara loga y = wx = loga b logb y .

1 loga y loga b

Askseic.1. log2 4, log 1 2, log 1 4. 2 2 2. log2 3 log3 4. 3. log2 3 log3 5 log5 7 log7 10 log10 8. 4. a > 0, a = 1. aloga y = y y > 0. 5. log2 3 ;1 6. a > 0, a = 1. log a y = loga y y > 0.

7. a > 0, a = 1. logaz (y z ) = loga y y > 0 z.

1.5 Trigwnometriko arijmo. Antstrofoi trigwnometriko arijmo. 1 , ( ) ( ). x |x|, , x > 0, ( ), x < 0. x , . 32

1. , 2 , , 3 2 . x 2 [0, 2] x 2 . x [k2, (k + 1)2], k : 2. , x, x 2.

1.4: .

. x , , 1. . cos x = + , , , . 2. . sin x = + , , , . 33

3. tan x = + , , , .

.

4. . cot x = + , , , . tan x , , , x = + . , cot x 2 , , , x = . (cos x, sin x) . cos x x, sin x x, tan x x cot x x. x. , . : (1) cos 0 = 1, sin 0 = 0, tan 0 = 0. cot 0. (2) cos = 0, sin = 1, cot = 0. tan . 2 2 2 2 (3) cos = 1, sin = 0, tan = 0. cot . (4) cos 3 = 0, sin 3 = 1, cot 3 = 0. tan 3 . 2 2 2 2 cos x > 0, x ( + k2, + k2), cos x < 0, x 2 2 ( +k2, 3 +k2), k . , sin x > 0, 2 2 x (k2, + k2), sin x < 0, x ( + k2, 2 + k2), k . , , 1 cos x 1, 1 sin x 1

x. . . 1.17 (1) (sin x)2 + (cos x)2 = 1. x sin (2) tan x = cos x , cot x = cos x . x sin (3) cos(x) = cos x, sin(x) = sin x, tan(x) = tan x, cot(x) = cot x. (4) cos( x) = sin x, sin( x) = cos x, tan( x) = cot x, cot( x) = tan x. 2 2 2 2 (5) cos(x + ) = cos x, sin(x + ) = sin x, tan(x + ) = tan x, cot(x + ) = cot x. (6) cos(x + y) = cos x cos y sin x sin y, sin(x + y) = sin x cos y + cos x sin y. 34

(7) cos x cos y = 2 sin xy sin x+y , sin x sin y = 2 sin xy cos x+y . 2 2 2 2 (8) k . (i) cos x > cos x , k2 x < x + k2, (ii) cos x < cos x , + k2 x < x 2 + k2. (9) k . (i) sin x < sin x , + k2 x < x + k2, 2 2 (ii) sin x > sin x , + k2 x < x 3 + k2. 2 2mkoc tou AD = mkoc tou EM kai, epomnwc, mkoc tou OA mkoc tou OE mkoc tou BG = mkoc tou ZM sin tan x = cos x . Epshc, sta moia trgwna OBG, OZM enai x mkoc tou OB mkoc tou OZ x kai, epomnwc, cot x = cos x . sin (3) Kai oi tsseric isthtec prokptoun ap to ti ta shmea tou kklou pou antistoiqon stouc x, x enai summetrik wc proc th dimetro A'OA. (4) Kai oi tsseric isthtec prokptoun ap to ti ta shmea tou kklou pou antistoiqon stouc x, x enai summetrik wc proc th diqotmo thc gwnac AOB. 2 (5) Kai oi tsseric isthtec prokptoun ap to ti ta shmea tou kklou pou antistoiqon stouc x, x + enai summetrik wc proc to shmeo O. (6) 'Estw M, N kai K ta shmea tou kklou pou antistoiqon stouc x, y kai x + y . Ta txa AK (aut pou periqei to M) kai NM (aut pou periqei to A) qoun to dio mkoc. Epomnwc, kai oi qordc AK kai NM qoun to dio mkoc, opte(2) Sta moia trgwna OAD, OEM enai (cos(x + y) 1)2 + (sin(x + y) 0)2 = (cos x cos(y))2 + (sin x sin(y))2 . (1) Efarmzoume to Pujagreio Jerhma sto trgwno OEM.

Gia thn apdeixh thc Prtashc 1.16 qrhsimopoiome aplc gewmetrikc nnoiec.

Knontac prxeic, qrhsimopointac tic (1) kai (3), prokptei h prth isthta sthn (6). H deterh isthta prokptei ap thn prth, qrhsimopointac tic (3) kai (4):sin(x + y) = = = (7) Enai cos x = cos x+y xy + 2 2 x+y xy 2 2 = cos = cos x+y xy x+y xy cos sin sin 2 2 2 2 x+y xy x+y xy cos + sin sin . 2 2 2 2 (x + y) = cos 2 x cos(y) sin cos 2 sin x cos y + cos x sin y. cos x + (y) 2 x sin(y) 2

kaicos y = cos

Epomnwc,

x+y xy sin . 2 2 Me ton dio trpo prokptei kai h deterh isthta thc (7). cos x cos y = 2 sin

(8) An k2 x < x + k2 , ta shmea M, M' tou kklou pou antistoiqon stouc x, x enai sto pnw hmikklio kai to M enai dexi tou M', opte cos x > cos x . An + k2 x < x 2 + k2 , tte ta dia shmea M, M' enai sto ktw hmikklio kai to M enai arister tou M', opte cos x < cos x . (9) An + k2 x < x + k2 , ta shmea M, M' tou kklou pou antistoiqon 2 2

stouc x, x enai sto dexi hmikklio kai to M enai ktw ap to M', opte sin x < sin x . An + k2 x < x 3 + k2 , ta dia shmea M, M' enai sto arister hmikklio kai to M 2 2 enai pnw ap to M', opte sin x > sin x .

. . 35

. 1. y [1, 1]. y . arccos y [0, ] , . 2. y [1, 1]. y . arcsin y [ , ] . 2 2 3. y. y. . arctan y ( , ) . 2 2 4. y. y. . arccot y (0, ) . : (1) arccos 1 = 0, arccos 0 = , arccos(1) = . 2 (2) arcsin 1 = , arcsin 0 = 0, arcsin(1) = . 2 2 (3) arctan 0 = 0 arccot 0 = . 2 arccos y - y, arcsin y - y, arctan y - y arccot y - y. y. y [1, 1] arccos y [0, ] cos x = y. x = arccos y cos x = y 0 x .

cos x = y [, 0], arccos y, : arccos y + k2 arccos y + k2, k . , , y [1, 1] arcsin y [ , ] sin x = y. 2 2 36

x = arcsin y

sin x = y

x . 2 2

sin x = y [ , 3 ], arcsin y, 2 2 : arcsin y + k2 arcsin y + k2, k . y arctan y ( , ) 2 2 tan x = y. x = arctan y tan x = y arccos y arcsin y < arcsin y . (2) y < y . arctan y < arctan y arccot y > arccot y .antstoiqa shmea tou hmikuklou A'BA. To E enai arister tou E', opte to M enai arister tou M' kai, epomnwc, enai arccos y > arccos y . Omowc, stw Z, Z' ta shmea thc diamtrou B'OB pou antistoiqon stouc y, y kai M, M' ta antstoiqa shmea tou hmikuklou B'AB. To Z enai ktw ap to Z', opte to M enai ktw ap to M' kai, epomnwc, enai arcsin y < arcsin y . (2) 'Estw D, D' ta shmea thc eujeac e pou antistoiqon stouc y, y kai M, M' ta antstoiqa shmea tou hmikuklou B'AB. To D enai ktw ap to D', opte to M enai ktw ap to M' kai, epomnwc, enai arctan y < arctan y . 'Estw G, G' ta shmea thc eujeac h pou antistoiqon stouc y, y kai M, M' ta antstoiqa shmea tou hmikuklou A'BA. To G enai arister tou G', opte to M enai arister tou M' kai, epomnwc, enai arccot y > arccot y .(1) 'Estw E, E' ta shmea thc diamtrou A'OA pou antistoiqon stouc y, y kai M, M' ta

Askseic. . 1. ( ) , . 6 4 3 2. . cos x = 1 , 2 1 sin x = , 2 cot x = 1, 37 1 3 cos x = , sin x = , 2 2 tan x = 3 , cot x = 3 .

tan x = 0,

3. |a cos x + b sin x|

a2 + b2 .

4. a, b a2 + b2 = 1 q [0, 2) cos q = a sin q = b. 5. a, b 0. , p > 0 q a cos x + b sin x = p cos(x q) x. (: a cos x + b sin x = 6. (i) cos y = cos x y = x + k2 y = x + k2, (ii) sin y = sin x y = x + k2 y = x + k2, (iii) tan y = tan x y = x + k, (iv) cot y = cot x y = x + k, k . 7. 1 + (tan x)2 = 8. tan(x + y) = 9. cos(2x) = (cos x)2 (sin x)2 = 2(cos x)2 1 = 1 2(sin x)2 , sin(2x) = 2 sin x cos x, tan(2x) = 10. cos x = tan x = 1 (tan x )2 2 , 1 + (tan x )2 2 2 tan x 2 , 1 (tan x )2 2 38 sin x = cot x = 2 tan x 2 , 1 + (tan x )2 2 1 (tan x )2 2 . 2 tan x 2 2 tan x , 1 (tan x)2 cot(2x) = (cot x)2 1 . 2 cot x tan x + tan y , 1 tan x tan y cot(x + y) = cot x cot y 1 . cot x + cot y 1 , (cos x)2 1 + (cot x)2 = 1 . (sin x)2 a2 + b2 a a2 +b2 b a2 +b2

cos x +

sin x .)

. 1. 0, 1 , 22 , 23 1. 2 2. arccos y + arcsin y = , 2 arctan y + arccot y = . 2

y [1, 1] y. 3. y y = cos(arccos y) y = sin(arcsin y); y y = tan(arctan y) y = cot(arccot y); 4. arccos(cos x) = x x [0, ]. arccos(cos x), x [k, (k + 1)], k = 0; arcsin(sin x), arctan(tan x) arccot(cot x);

39

40

Keflaio 2

Akoloujec kai 'Oria Akoloujin . : - . n0 . . : , e .

2.1 Orismo. ( ) : , , . . : x1 , x2 , . . . , xn , . . . , y1 , y2 , . . . , yn , . . . , z1 , z2 , . . . , zn , . . . .

(xn ), (yn ), (zn ).

n ( , ) , : , 41

. xn+1 xn xn1 xn .1 1 : (1) ( n ), 1, 1 , 1 , . . . , n , . . . . 2 3 (2) (n), 1, 2, 3, 4, . . . , n, . . . . (3) (1), 1, 1, 1, . . . , 1, . . . . (4) (1)n1 , 1, 1, 1, 1, . . . , 1, 1, . . . . 1 1 1 1 1 (5) 10n , 10 , 102 , 103 , . . . , 10n , . . . . (6) n- n, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, . . . .

(xn ) , xn+1 xn n. (xn ) , xn+1 > xn n. (xn ) , xn+1 xn n. , (xn ) , xn+1 < xn n. , , , . : , . (xn ) , , , c xn = c n. . . : (1) (, ) . , , ( , ). (2) , . {1} . , , . 1, 1, 1, . . . . , , (, , ) ( ). . . (xn ) , , n , xn . , (xn ) , xn . n ( , ) 42

. , .

2.1: . (xn ) , , 0. n xn . (n, xn ) (xn ). , xn , (n, xn ) . , : n , (n, xn ) . , (xn ) , , n , (n, xn ) ( ). , (xn ) , (n, xn ) ( ). , (xn ) , (n, xn ) .

2.2: .

43

Askseic.1. . 2n 1 , 3n + 2 (2n! ), n , n+1 (1)n+1 , n! 1 (1)n , n3 1 1 1 1 + + + + n , 2 4 8 2 n2 3n + 1 , 2n n!

(1)n1 , 2 4 6 2n

(2x)n1 , (2n 1)5

(1)n1 x2n1 , (2n 1)!

(1)n x2n1 . 1 3 5 (2n 1)

2. : 1, 4, 9, 16, 25. , ; (i) 36. (ii) 24. (iii) . 3. . (n), (n), (1)n1 , 1 + (1)n1 , 2 a+b ab + (1)n1 . 2 2

4. n 2[ n ] n 3[ n ] . 2 3 , m , n n m[ m ] . 5. (xn ) , . 6. (xn ) (xn ) , . 7. ; ; (n), (1)n1 , (1)n1 n , n + 15 16 (1)n1 , n 8n , n! 2 n 2 , 2n , 1 , 2n n 3 .

8n 1 , 2+n+1 n

,

n3

, , . . 44

8. . a, b, p, q, p, q 0. (xn ) : x1 = a, x2 = b xn+2 = pxn+1 + qxn (n 1).

n- xn . 1: p = 0, q = 0. xn = bpn2 n 2. 2: p = 0, q = 0. xn = aq n2 , xn = bq 2 , n .n1 2

, n

3: p = 0, q = 0. x2 = px + q. (i) = p2 + 4q > () , 0, 1 = p+2 2 = p2 . , + = a 1 + 2 = b . xn = 1 n1 + 2 n1 n 1. (ii) = p2 + 4q = 0, , = p . 2 , = a + = b . xn = n1 + (n 1)n1 n 1. (iii) = p2 + 4q < 0 ( q < 0), () , 1 = p+i 2 2 = pi 2 . p 2 2 = q > 0 2 + 2 = 1, p [0, 2) cos = 2 sin = . 1 = (cos + i sin ) 2 = (cos i sin ). 2 cos(2) = p cos + q 2 sin(2) = p sin . , = a ( cos + sin ) = b . , xn = n1 cos((n 1)) + sin((n 1)) n 1. n- ( ) x1 = x2 = 1 45 2

. xn+2 = 3xn , xn+2 = 2xn+1 xn , xn+2 = xn+1 + xn , xn+2 = xn+1 xn .

, x1 = x2 = 1 xn+2 = xn+1 + xn , Fibonnaci 1, 1, 2, 3, 5, 8, 13.

2.2 'Orio akoloujac. : 1 n . 1 > 0, , n0 n0 , n01 , n01 , . . . +1 +2 1 < . n 0 1 1 n , ( n ) 0 . 1 n0 n < ; 1 1 n < n > (, n) (i) a 0, n0 = [a]+1 > a (ii) a < 0, n0 = 1 > a. n0 n0 = [ 1 ] + 1. : ( n1 ), n 1 2 3 4 5 6 99 100 99999 , , , , , ,..., , ,..., , ... . 2 3 4 5 6 7 100 101 100000 , n, n 1, n , ( 1) . , n1 1. , n 1 1 n1 1 | n1 1| = | n | = n . n n n1 ( n ) 1 . 0, : .(1)n1 n (1) nn1

, 1, 1 , 1 , 1 , . . . . 2 3 4(1)n1 n

n- 0

0 =

1 n

.

0

, ( 46

0). , 0, 1 0 . . : (xn ), xn = xn =2 n 1 n 3+(1)n 2n

, ,

, n , , n .

1 1, 1, 1 , 1 , 1 , 1 , 1 , 4 , . . . . 3 2 5 3 7 xn 0 ( xn ) , , : .

, . (xn ) x x x (xn ), |xn x| . (xn ) x xn x lim xn = x n1

n+

lim xn = x.n

1 : lim n = 0, lim n1 = 1, lim (1) n n

= 0 lim 3+(1) = 0. 2n

: lim xn = x, |xn x| , > 0 n0 |xn0 x|, |xn0 +1 x|, |xn0 +2 x|, . . . < . , > 0 n0 |xn x| < n n0 .

2.3: x < xn < x + n n0 . lim xn = x ; (xn ) x xn x , , xn x. : (c), c, c, c, c, . . . . lim c = c. 47

, c |c c| = 0 , , ( , n0 = 1 ).1 : ( n ). a > 0 1 1 1 ( na ), 1, 2a , 3a , 41 , . . . . a

lim

1 =0 na

(a > 0).

1 > 0 | na 0| < 1 1 n > ( 1 ) a . , , n0 = [( 1 ) a ] + 1, 1 1 | na 0| < n n0 . | na 0| 1 , ( na ) 0.

: . x 0, XN . . . X0 , x1 x2 x3 . . . (sn ) . 1.3. (sn ) xsn . , |sn x| = xsn , , lim sn = lim XN 10N + + X1 10 + X0 + x1 xn + + n 10 10 = x.

1 1.3. tn = sn + 10n (tn ) |tn x| = tn x .

lim tn = x.1 , |sn x| = x sn 10n |tn x| = tn x , |sn x| |tn x| 1 > 0 10n < . 1 n > log10 log10 1 0, 1, log10 1 < 0, > 1. , n0 = [log10 1 ] + 1, 1, n0 = 1, > 1, 1 10n < |sn x| < |tn x| < n n0 . 1 10n

: . (1)n1 , 1, 1, 1, 1, . . . , 1, 1, . . . , . ( ) : 1 1 . (xn ) , (xn ) . : . (an ), 48

a, a2 , a3 , a4 , . . . . a. a = 1, (1) , , 1. , a = 0, (0) 0. a 1 ( : a = 1 ), , . , a 1, a2 1, a3 1, a4 1, . . . 1 1 . 0 < |a| < 1, 0. 1 1 a = 2 a = 10 . a = 1 , 2 1 1 1 1 1 1 1 1 1 2 , 22 , 23 , 24 , . . . , a = 10 , 10 , 102 , 103 , 104 , . . . . 0. , , 0 < |a| < 1 . |an 0| < |a|n < n > log|a| . , log|a| 0, 1, log|a| < 0, > 1. , n0 = [log|a| ] + 1, 1, n0 = 1, > 1, |an 0| < n n0 . , |an 0| , (an ) 0. a > 1. .

Askseic.1. . lim 3 , 4 lim 1 n1 3

,

1 lim , n n

lim

3n , 4n

lim

(1)n 2n . 32n

2. : n2 1 lim = , 3n + 4 3 3n lim = 2, n+3 n lim = 0, n+1 n+1 = 1. lim 2 n+1

; , n- . n0 . 3. 3n1 4n+5 , , , , . . 4. lim 3n1 = 3 . 4n+5 4 > 0 3n1 3 < . 4n+5 4 n0 . 49

3n1 3 < , , 4n+5 4 n. n . n0 n n0 . : (i) . (ii) n0 . (iii) n0 . n0 , 19 > 20 , , 0 < 19 . 20 5. . n2 n + 1 1 2 n+3 2 n + 2n + 1 = , lim lim = , lim = 1. 3n2 + 2 3 3 43 n 2n + 3 n + 4 6. (n3[ n ]) 3 3 4 .3 2

, (xn ) N M , N < M . (xn ) () ; 7. (xn ) . (xn ) ; ; (xn ) ;

2.3 Ta wc ria akoloujin. : 1.3 n, , . M > 0 n n, n+1, n+2, . . . > M . , n0 = [M ] + 1, n > M n n0 . , , (n) . : ( n) M . n > M n > M 2 . n , M 2 , n M . , n0 = [M 2 ] + 1, n > M 2 , , n > M n n0 . M , n, ( n), . : 50

(n), ( n) .

, . (xn ) + + + (xn ), xn . (xn ) + xn + lim xn = + n+

lim xn = + .

, (xn ) (xn ), xn . (xn ) xn lim xn = n+

lim xn = .

: (1) lim = +, lim n (2) lim(n) = , lim( n) = .

n = +.

: lim xn = +, xn , M > 0 n0 xn0 , xn0 +1 , xn0 +2 , . . . > M . , M > 0 n0 xn > M n n0 . lim xn = , xn , M > 0 n0 xn0 , xn0 +1 , xn0 +2 , . . . < M . , M > 0 n0 xn < M n n0 .

2.4: xn > M n n0 . lim xn = + lim xn = ; (xn ) + xn . , + () , (xn ) + xn () +. , (xn ) xn xn () . 51

: + () () . , + . , ( ) . +, . + . , . : (xn ), xn = 1+(1) n + 1(1) n2 , 2 2 , n, n , xn = n2 , n . , , 1, 2, 9, 4, 25, 6, 49, 8, . . . . lim xn = +, xn . (xn ) : xn . : (1)n1 . + . : , +, . : (n) ( n). a > 0 (na ), 1, 2a , 3a , 4a , . . . . lim na = + (a > 0).n n

M 1 na > M n > M a . , n0 = 1 a [M a ] + 1, n > M n n0 . na (na ) +. : a > 1. (loga n), loga 1 = 0, loga 2, loga 3, loga 4, . . . . lim loga n = + (a > 1).

, M loga n > M n > aM . n0 = [aM ] + 1, loga n > M n n0 . loga n M > 0, lim loga n = +. 52

: . a > 1 (an ). +. a = 2 a = 10 2, 22 , 23 , 24 , . . . 10, 102 , 103 , 104 . . . . +. , , M > 0 an > M n > loga M . , loga M 0, M 1, loga M < 0, M < 1. , n0 = [loga M ] + 1, M 1, n0 = 1, M < 1, an > M n n0 . an M > 0 (an ) +. , a 1, (an ) , 1 1 . , , + , , . : = +, a > 1, = 1, a = 1, n lim a 1 < a < 1, = 0, , a 1. : (1)n1 n , 1, 2, 3, 4, 5, 6, 7, . . . , . . : , lim, limn+ : . : , .

Askseic.1. , , . lim(n n), n2 lim , n lim log10 n, lim 22n , 3n lim(3)2n ,n

lim(3)3n .

2. x = 1 lim (1x)n . (1+x) 3. (;) +. (n2 18n4), 7n 1 2 n , 30 n , 30 n + 1 53 1n , 1+ n n2 + 1 . n + 100

4. n n. n2 2n 106 1, n2 2n 106 , n + (1)n > 103 .

5. n n . (1)n1 n > 1000,2

1 + (1)n1 n < 2.

6. lim n 5 = +. 2n+1 M > 0 n 5 > M . 2n+1 n0 . M < n 5 , M , 2n+1 n. n . n0 n n0 . : (i) M M M . (ii) n0 M . (iii) n0 . n0 M M . 7. . n+3 lim = +, 3 n+4 lim(2n n2 ) = , lim 3n4 + 2n2 = +, 4n2 + 1 n 2n = . lim n+12 2

M > 0 n2n 2n n2 < M n+1 < M , .

2.4 Idithtec sqetikc me ria akoloujin.. . (xn ) , l, u l xn u n. : (1) (c) , , . [c, c]. 54

1 (2) ( n ) [0, 1].

(3) (1) [ 1 , 1]. n 2 n1 (4) ( n ) [0, 1]. (5) (1)n1 [1, 1]. (6) (1)n1 n + n , 2, 0, 6, 0, 10, 0, 14, 0, . . . , . (7) 2, 0, 6, 0, 10, 0, 14, 0, . . . , , . (8) (1)n1 n , 1, 2, 3, 4, 5, 6, . . . , . : (xn ) , xn [l, u]. (xn ) 0. [M, M ] [l, u]. , , , (xn ) , M |xn | M n.1 ( n ), (1) ( n1 ) , n n , . ( ) .n1

n1

2.1 , .'Estw ti lim xn = x. Epeid oi apostseic |xn x| gnontai mikrterec ap kje jetik arijm - gia pardeigma, ton = 1 - uprqei fusikc n0 ste na enai |xn x| < 1 gia kje n n0 . Ap thn |xn x| < 1 sunepgetai |xn | = |(xn x) + x| |xn x| + |x| < 1 + |x|. 'Ara enai |xn | < 1 + |x| gia kje n n0 . Orzoume M = max{|x1 |, . . . , |xn0 1 |, 1 + |x|} kai me lgh skyh blpoume ti enai |xn | M gia kje n.

:

2.1.

: (1)n1 . (xn ) , , u xn u n. , (xn ) , , l l xn n. . . 2.2 (1) +, . (2) , .(1) 'Estw lim xn = +. Epeid oi xn gnontai megalteroi ap kje jetik arijm - gia pardeigma, ton M = 1 - uprqei fusikc n0 ste na enai xn > 1 gia kje n n0 . Orzoume l = min{x1 , . . . , xn0 1 , 1} kai ekola blpoume ti enai xn l gia kje n. Ap thn llh

55

meri, h (xn ) den qei kanna nw frgma afo oi xn gnontai megalteroi ap kje jetik arijm. (2) H apdeixh, me tic profanec allagc, enai parmoia me thn apdeixh tou (1).

: (1), (2) 2.2 . : (1) n+(1) n , 1, 0, 3, 0, 5, 0, 7, . . . . n2 0, n , n, n . 0, . , , , . , +. , (: ) . 0 1, . (2) 1, 0, 3, 0, 5, 0, 7, . . . , . . .n1

: (+) = , () = + .

: + , (+) , . (+) . () +. (xn ) (xn ), n- n- . 2.3 . (xn ) , lim(xn ) = lim xn .'Estw lim xn = x. Parnoume opoiondpote > 0, opte uprqei fusikc n0 ste na enai |xn x| < gia kje n n0 . Parathrome ti isqei |(xn ) (x)| = |x xn | = |xn x|. 'Ara enai |(xn ) (x)| < gia kje n n0 . 'Ara oi apostseic |(xn ) (x)| gnontai mikrterec ap kje jetik arijm kai, epomnwc, lim(xn ) = x = lim xn . 'Estw lim xn = +. Parnoume opoiondpote M > 0, opte uprqei fusikc n0 ste na enai xn > M gia kje n n0 . Sunepgetai xn < M gia kje n n0 . 'Ara oi xn gnontai mikrteroi ap kje arnhtik arijm kai, epomnwc, lim(xn ) = = (+) = lim xn . Omowc, apodeiknetai ti, an lim xn = , tte lim(xn ) = + = () = lim xn .

: (+) + x = +, x + (+) = +, 56 (+) + (+) = +,

() + x = , ,

x + () = ,

() + () = .

(+) + (),

() + (+) .

: (+) + (+) + : , , . (+) + x + : x , , . . (+) + () . , : . (xn ), (yn ) (xn + yn ) n- n- . 2.4 . lim xn , lim yn lim xn + lim yn , lim(xn + yn ) = lim xn + lim yn .'Estw lim xn = x kai lim yn = y . Parnoume opoiondpote > 0, opte uprqei fusikc n0 ste na enai |xn x| < 2 gia kje n n0 kai fusikc n0 ste na enai |yn y| < 2 gia kje n n0 . Orzoume ton fusik n0 = max{n0 , n0 }, opte enai n0 n0 kai n0 n0 . 'Ara enai |xn x| < 2 kai |yn y| < 2 gia kje n n0 . Parathrome ti isqei |(xn + yn ) (x + y)| = |(xn x) + (yn y)| |xn x| + |yn y|. Sunepgetai ti enai |(xn +yn )(x+y)| < 2 + 2 = gia kje n n0 kai, epomnwc, oi apostseic |(xn +yn )(x+y)| gnontai mikrterec ap kje jetik arijm. Dhlad lim(xn + yn ) = x + y = lim xn + lim yn . 'Estw lim xn = + kai lim yn = + lim yn = y . Tte h (yn ) enai ktw fragmnh, dhlad uprqei l ste na enai yn l gia kje n. Parnoume opoiondpote M > 0, opte, epeid lim xn = +, uprqei fusikc n0 ste na enai xn > M l gia kje n n0 . Sunepgetai xn + yn > (M l) + l = M gia kje n n0 . 'Ara oi xn + yn gnontai megalteroi ap kje jetik arijm, opte lim(xn + yn ) = + = (+) + (+) = lim xn + lim yn lim(xn + yn ) = + = (+) + y = lim xn + lim yn . Oi uploipec periptseic qoun parmoia aitiolghsh.1 1 : (1) lim n + (1) = 0 + 0 = 0. = lim n + lim (1) n n 2 n +1 1 1 (2) lim n = lim(n + n ) = lim n + lim n = (+) + 0 = +. (3) lim(n n) = lim(n) + lim( n) = () + () = .n1 n1

57

+, : : (1) lim(n + c) = lim n + lim c = (+) + c = +, lim(n) = lim (n + c) + (n) = lim c = c. (2) lim(2n) = +, lim(n) = lim (2n) + (n) = lim n = +. (3) lim n = +, lim(2n) = lim (n) + (2n) = lim(n) = . (4) n + (1)n1 n 1 n lim(n 1) = +. , lim n + (1)n1 = +. , lim(n) = lim n + (1)n1 + (n) = lim(1)n1 . (+) x = +, () x = , x () = +, x (+) = , (+) () = +, () (+) = ,

. (+) (+), () () .

: + () ( ) () x x. (xn ), (yn ), (xn yn ), , , , xn yn = xn + (yn ). 2.5 . lim xn , lim yn lim xn lim yn , lim(xn yn ) = lim xn lim yn . . () x = , x > 0, () x = x < 0 () () = + , () 0, 58 0 () () ( ) = . , x () = , x () = ,

.

: (+) (+) = + : , , . (+) x = + ( x > 0) : x , , . . (+) 0 , 0 : . (xn ), (yn ) (xn yn ) n- n- . 2.6 . lim xn , lim yn lim xn lim yn , lim(xn yn ) = lim xn lim yn .'Estw lim xn = x kai lim yn = y . Tte h (yn ) enai fragmnh, dhlad uprqei kpoioc M ste na enai |yn | M gia kje n. Parnoume opoiondpote > 0, opte uprqei fusikc n0 ste na enai |xn x| < 2M +1 gia kje n n0 kai fusikc n0 ste na enai |yn y| < 2|x|+1 gia kje n n0 . Orzoume ton fusik n0 = max{n0 , n0 }, opte enai n0 n0 kai n0 n0 . 'Ara enai |xn x| < 2M +1 kai |yn y| < 2|x|+1 gia kje n n0 . Parathrome ti isqei |xn yn xy| = |(xn x)yn +x(yn y)| |xn x||yn |+|x||yn y| M |xn x|+|x||yn y|. Sunepgetai ti enai |xn yn xy| M 2M +1 + |x| 2|x|+1 < 2 + 2 = gia kje n n0 kai, epomnwc, oi apostseic |xn yn xy| gnontai mikrterec ap kje jetik arijm. Dhlad lim(xn yn ) = xy = lim xn lim yn . 'Estw lim xn = + kai lim yn = +. Parnoume opoiondpote M > 0, opte uprqei fusikc n0 ste na enai xn > M gia kje n n0 kai fusikc n0 ste na enai yn > M gia kje n n0 . Orzoume ton fusik n0 = max{n0 , n0 }, opte enai n0 n0 kai n0 n0 . 'Ara enai xn > M kai yn > M gia kje n n0 . Sunepgetai xn yn > M M = M gia kje n n0 . 'Ara oi xn yn gnontai megalteroi ap kje jetik arijm, opte lim(xn yn ) = + = (+)(+) = lim xn lim yn . 'Estw lim xn = + kai to lim yn = y enai jetikc arijmc. Parnoume opoiondpote M > 0, opte uprqei fusikc n0 ste na enai xn > 2M gia kje n n0 kai fusikc n0 y ste na enai |yn y| < y gia kje n n0 . Orzoume ton fusik n0 = max{n0 , n0 }, 2 opte enai n0 n0 kai n0 n0 . 'Ara enai xn > 2M kai |yn y| < y gia kje n n0 . y 2 Apo thn |yn y| < y sunepgetai yn > y y = y gia kje n n0 . Sunepgetai xn yn > 2 2 2 2M y = M gia kje n n0 . 'Ara oi xn yn gnontai megalteroi ap kje jetik arijm, opte y 2 lim(xn yn ) = + = (+)y = lim xn lim yn . 'Olec oi uploipec periptseic qoun ousiastik thn dia aitiolghsh. Oi mnec allagc qoun na knoun me ton gnwst kanna ginomnou prosmwn.

59

1 1 : (1) lim (1)2 = lim n (1) = lim n lim (1) = n n n 0 0 = 0. 1 1 (2) lim n1 = lim( n1 n ) = lim n1 lim n = 1 0 = 0. n2 n n 2 (3) lim(n n ) = lim n(1 n) = lim n lim(1 n) = lim n (lim 1 lim n) = (+) (1 (+)) = (+) () = .

n1

n1

n1

: . (xn ) c = 0. lim xn , lim(cxn ) = c lim xn . , (xn ) (c) c. : a > 0. c > 0, lim(cna ) = c lim na = c (+) = +. c < 0, lim(cna ) = c lim na = c (+) = . : a > 0, lim(cna ) = c lim na = c 0 = 0. : n. a0 + a1 x + + aN xN ( aN = 0 N 1). lim a0 + a1 n + + aN nN = aN (+) = +, aN > 0, , aN < 0.

, aN nN , a0 + a1 n + + aN nN = aN nN a0 1 a1 1 aN 1 1 + + + +1 , a N nN aN nN 1 aN n

1, 0. lim(a0 + a1 n + + aN nN ) = lim(aN nN ) 1 = aN (+). n . lim a0 + a1 n + + aN nN = lim(aN nN ) . : lim(3n2 5n + 2) = lim(3n2 ) = +, lim( 1 n5 + 4n4 n3 ) = 2 1 lim( 2 n5 ) = . : . a (1 + a + a2 + + an1 + an ), 1 + a, 1 + a + a2 , 1 + a + a2 + a3 , . . . . : lim(1 + a + a2 + + an ) = +, a 1, 1 = 1a , 1 < a < 1, , a 1. 60

. n+1 1 1 a > 1, lim(1 + a + a2 + + an ) = lim a a1 = a1 lim(aan 1) = 1 a1 (a (+) 1) = +. a = 1, lim(1 + a + a2 + + an ) = lim(n + 1) = +. n+1 1 < a < 1, lim(1 + a + a2 + + an ) = lim 1a = 1a 1 1 1 lim(1 aan ) = 1a (1 a 0) = 1a . 1a , a 1. an+1 1 = (a 1)(1 + a + a2 + + an ) , lim(1 + a + a2 + + an ), lim(an+1 1) = lim(a1)(1+a+a2 + +an ) = (a1) lim(1+a+a2 + +an ). 1 1 , , lim an = lim a (an+1 1)+1 = a lim(an+1 1)+1 . n 2 lim a , lim(1 + a + a + + an ) . , ()k k : (+)k = (+) (+) = +,k k

()k = () () = .k

() = +, k , , k . . 2.7 lim xn k , lim xn k = (lim xn )k . lim xn k = lim xn xn = lim xn lim xn = (lim xn )k .k k

: (1) lim( n1 )3 = (lim n1 )3 = 13 = 1. n n 8 (2) lim(n5 2n2 + n 7)8 = lim(n5 2n2 + n 7) = (+)8 = +. (3) lim(2n3 + n2 + 2n 7)4 = lim(2n3 + n2 + 2n 7) (4) lim(n3 + 2n 1)5 = lim(n3 + 2n 1)5 4

= ()4 = +.

= ()5 = .

+, 0, .1 1 : (1) lim(cn) = + (c > 0), lim n = 0 lim(cn n ) = lim c = c. 1 1 (2) lim(cn) = + (c > 0), lim( n ) = 0 lim cn ( n ) = lim(c) = c. 1 2 2 1 (3) lim n = +, lim n = 0 lim(n n ) = lim n = +. 1 1 1 (4) lim n = +, lim n2 = 0 lim(n n2 ) = lim n = 0.

(5) lim n = +, lim (1) n .

n1

= 0 lim n

(1)n1 n

= lim(1)n1

1 = 0, + 61 1 = 0.

1 0

.

1 : = 0 : , , ( 0) . 1 . 0 0 , . , , , . , , .

(xn ) ( x1 ) , , n xn = 0 n. 2.8 , . xn = 0 n. 1 lim xn lim xn ( lim xn = 0), lim 1 1 = . xn lim xn

'Estw ti to lim xn = x enai jetikc arijmc. Parnoume opoiondpote > 0, opte 2 uprqei fusikc n0 ste na enai |xn x| < min{ x2 , x } gia kje n n0 . Sunepgetai 2|xn x| < xn > x x 2 x2 2

kai |xn x| 0, opte uprqei fusikc n0 ste na enai xn > 1 gia kje n n0 . Sunepgetai 0 < x1 < kai, epomnwc, x1 0 = x1 < n n n 1 1 gia kje n n0 . 'Ara lim x1 = 0 = + = lim xn . n H aitiolghsh tou kanna stic periptseic pou to rio enai arnhtikc arijmc enai parmoia.

: a > 1, lim log1

a

n

=

1 lim loga n

=

1 +

= 0.

: n1 n 0. , lim (1) = 0, lim (1)n1 = lim(1)n1 n n . . 62

2.8. , . . 2.9 , . (1) lim xn = 0 (xn ) , lim x1 = +. n (2) lim xn = 0 (xn ) , lim x1 = . n(1) 'Estw lim xn = 0 kai xn > 0 gia kje n. Parnoume opoiondpote M > 0, opte uprqei 1 fusikc n0 ste na enai 0 < xn < M gia kje n n0 . Sunepgetai x1 > M gia kje n 1 n n0 , opte oi x gnontai megalteroi ap kje jetik arijm. 'Ara lim x1 = +. n n (2) H apdeixh enai parmoia me thn apdeixh tou (1).

: = , x x > 0, x < 0, = x ,

x = 0.

x , , , 0 0 . : + () ( ) () x x. (xn ) (yn ), n ( xn ), y n , xn = xn y1 . y n 2.10 . yn = 0 n. n lim xn , lim yn lim xn , lim y lim xn lim xn = . yn lim yn

: n. N 0 +a x++aN x a0 +b11x++bM xM , aN = 0, bM = 0. b a N bM (+), N > M , a0 + a1 n + + aN nN aN lim = bM , N = M , b0 + b1 n + + bM nM 0, N < M . 63

, a0 + a1 n + + aN nN aN N M = n b0 + b1 n + + bM nM bMa0 1 aN nN b0 1 bM nM

+ +

a1 1 aN nN 1 b1 1 bM nM 1

+ + + +

aN 1 1 aN n + 1 bM 1 1 bM n + 1

, , N aN 1 0 +a n++aN n 1. lim a0 +b11n++bM nM = bM lim nN M 1 = b aN N M . , bM lim n , n . lim3

a0 + a1 n + + aN nN aN nN = lim . M b0 + b1 n + + bM n bM nM2 3 2 2

2n +n+1 n +2n+3 : lim n 2n2 3n1 = lim 2n2 = +, lim n n+2 = lim n = n

, lim

n4 4n3 +n2 n7 5 2n4 +n2 +n+1

n = lim 2n4 =

4

1 2

, lim n3n 2+n+4 = lim n = 0. +n +5n+6 n3+3n+1 lim n2n+32

2

2

: (1) lim (2) limn +n+7 3n3 +n2 +13

3

=

n2 +3n+1 7 = 2n+3 3 n3 +n+7 lim 3n3 +n2 +1

7

= ()7 = .

1 = ( 1 )3 = 27 . 3 0 0

, c 1 : (1) lim n = 0, lim n = 0 lim 1 1 (2) lim n = 0, lim n2 = 0 lim1 n 1 n2 1 n2 1 n c n 1 n

+ +

.

= lim c = c.

= lim n = +.

(3) (4) (5) (6)

1 1 1 lim n2 = 0, lim n = 0 lim = lim n = 0. lim(cn) = + (c > 0), lim n = + lim cn = lim c = c. n 2 lim n2 = +, lim n = + lim n = lim n = +. n n 1 lim n = +, lim n2 = + lim n2 = lim n = 0.

| + | = +, | | = +.

: | | = + : , , . , , (|xn |) (xn ). 2.11 (xn ) , lim |xn | = | lim xn |. 64

'Estw lim xn = x. Parnoume opoiondpote > 0, opte uprqei fusikc n0 ste na enai |xn x| < gia kje n n0 . Parathrome ti isqei |xn ||x| |xn x|, opte sunepgetai gia kje n n0 . 'Ara lim |xn | = |x| = | lim xn |. 'Estw lim xn = + lim xn = . Parnoume opoiondpote M > 0, opte uprqei fusikc n0 ste na enai xn > M xn < M , antistoqwc, gia kje n n0 . Kai stic duo periptseic sunepgetai |xn | > M gia kje n n0 . 'Ara oi |xn | gnontai megalteroi ap kje jetik arijm, opte lim |xn | = + = | | = | lim xn |.|xn | |x| 0. Epeid lim xn = +, uprqei fusikc n0 ste na enai xn > M gia kje n n0 kai, epeid yn xn , sunepgetai yn > M gia kje n n0 . 'Ara oi yn gnontai megalteroi ap kje jetik arijm, opte lim yn = +. (2) H apdeixh enai parmoia me thn apdeixh tou (1).

: (1) n+(1)n1 n1 n. lim(n1) = +, lim n + (1)n1 = +. 2 (2) n +2n+1 n lim n = +, n+2 lim n2

+2n+1 n+2

= +.

2.13 xn yn n. lim xn = x lim yn = y, x y.Ac upojsoume (gia na katalxoume se antfash) ti y < x. Parnontac = xy > 0, 2 ap to ti lim xn = x kai lim yn = y sunepgetai ti uprqei fusikc n0 ste na enai |xn x| < xy gia kje n n0 kai fusikc n0 ste na enai |yn y| < xy gia kje 2 2 n n0 . Orzoume n0 = max{n0 , n0 }, opte isqei n0 n0 kai n0 n0 . Sunepgetai ti enai |xn x| < xy kai |yn y| < xy gia kje n n0 . 'Ara xn > x xy = x+y kai 2 2 2 2 yn < y + xy = x+y gia kje n n0 . Dhlad enai xn > x+y > yn gia kje n n0 kai 2 2 2 aut antifskei me to ti xn yn gia kje n.

: (1) xn a n lim xn = x, x a. , (a) , a xn n lim a = a lim xn = x, a x. (2) xn b n lim xn = x, x b. 65

(1). (3) (xn ) [a, b] lim xn = x, x [a, b]. (1) (2). 2.14 . xn yn zn n. lim xn = lim zn = , lim yn = .Parnoume opoiondpote > 0, opte uprqei fusikc n0 ste na enai |xn | < gia kje n n0 kai fusikc n0 ste na enai |zn | < gia kje n n0 . Orzoume n0 = max{n0 , n0 }, opte isqei n0 n0 kai n0 n0 . Sunepgetai ti enai |xn | < kai |zn | < gia kje n n0 . 'Ara xn > kai zn < + gia kje n n0 . Sunepgetai < xn yn zn < + kai, epomnwc, |yn | < gia kje n n0 . 'Ara oi apostseic |yn | gnontai mikrterec ap kje jetik arijm, opte lim yn = .

: (1) 1 lim(1 = 1 lim(1 + 1 1 (2) n sin n n n sin n lim n = 0.1 n) 1 n)

1 n n. (1)n1 = 1, lim 1 + = 1. n 1 1 n. lim( n ) = lim n = 0, 1 n

1+

(1)n1 n

1+

Askseic. . 1. ; ; ; ( 2.1.7.) (n), (1)n1 , (1)n1 n , n + 15 16 , (1)n1 , n 8n , n! 2 n 2 2n , , 1 , 2n n3 n 3 .

8n 1 , 2+n+1 n

2. . 3. (xn ) (xn ) , . .

1. , , , , . lim 2n3 + 3n + lim 1 + 1 n3

1 , n

lim n +

1 (1)n1 +2 , n n9

lim

n2 n + 3 , n

,

lim 1 +

(1)n1 n 66

,

lim

(n + 1)7 (n + 2)3 (n + 3)79 , n89

lim

1 + 2+

1 n2 (1)n1 n

,

lim

n + 2+

1 n

(1)n1 n

,

1 + (1) n , lim n + 3 log10 n 1(1)n1 n2

n1

lim

n + (1)n n , 2 + (1)n lim 1 + 2 10n , 5 + 3 10n

lim

1 n

1 1 , + n2

lim

1 n

+

,

lim

1 n

1 1 , 2n2

lim

3n+1

3n + (2)n + (2)n+1

lim

log2 n + 3 . 2 log10 n + 15

2. n. lim(3n2 4n + 5), lim lim lim(n2 4n5 + 1), lim lim (1 n)5 + n4 7 , lim 3n2 + 4n , 2n 14

3n2 5n , 5n2 + 2n 64

2n5 + 4n2 , 3n7 + n3 103

2n 3 3n + 7 lim

,

lim

n2 + n + 1 3n + 1 lim

,

lim

n2 + n + 1 3n + 1

,

n2 n1 , n+1

n(n + 1) 4n3 2 . n+4 4n + 1

3. , , . lim(1 + 2 + 22 + + 2n ), lim 1 2 + 22 + + (1)n 2n , lim 1 + 1 1 1 + + + n , 2 22 2

lim 1 1 + 1 1 + + (1)n , lim 27 28 2n+6 + 8 + + n+6 , 7 3 3 3 lim 1 + 2 + + 2n , 1 + 3 + + 3n

lim lim

2n 2n+1 22n + n+1 + + 2n , n 3 3 3

1 3 + + (1)n 3n . 1 2 + + (1)n 2n

4. ; lim lim n n 3 3 , lim n + (1)n1 n , lim 1 + lim lim (1)n1 + 10 , n3 n , n+11 n2

1 + (1)n1 n , n

2 + (1)n1 n , n1 1+log3 n

lim(1)n1 lim 1(1)n1 n n

lim(1)n1 log2 n, lim 2(1)n1

1 (1)n1 +n1

,

+ .

,

,

lim 2(1)

n

,

lim 1 +

(1)n1 2

67

5. , . n5 + 4n3 < 100 n ; n7 35n6 + n3 47n < 84 n; 3 2

. . 7. : s. , , 12 . , ( ) , 48 . , (i) . (ii) . . 1. lim xn . (i) 1 < xn (ii)n2 +3n n2 +1

n.3+n 1+2n 3

log10 n2 2 log10 n+4

< xn 0, x = 0, lim[nx] = 0 , , x < 0. (: [a] a < [a] + 1.) lim([nx] [ny]) = +, 0, , x > y, x = y, x < y.

4. . lim 22n+(1)n1

n

= 0,

lim

1 (1)n1 + 2 4

n

= 0.

5. lim 1 n2 +1 + 1 n2 +2 + + n2 1 1 + = 1. 2+n +n1 n k 1 k n.)

(:

1 n2 +n

1 n2 +k

1 n2 +1

6. Fibonnaci (xn ), x1 = x2 = 1 xn+2 = xn+1 + xn . xn n n. 2 (xn ); 7. 2n n + 1 n n lim(3n 105 n) = +, lim n = 0. 3 7n 6n

1 n + 1 n 6 lim(7n n5n ) = +, lim n5n = 0. 7n

8. , (xn ) [a, b] lim xn = x, x [a, b]. x (xn ), (a, b); (: (0, 2) 1 1 ( n ) (2 n ).) x (xn ), (a, b); 9. x (rn ) rn < x n lim rn = x. (: rn x ;) 691 n

< rn < x.

2.5 Montonec akoloujec. O arijmo e, . (xn ) , xn ( n ) . . (xn ) - , (n) (n2 ) - xn . (xn ) - , ( n1 ) - xn , n , , . (xn ) +. , , xn (xn ). , , (xn ).

2.5: .

2.6: . (xn ). xn . (xn ) , xn , (xn ) , xn , . (xn ) xn (xn ) (xn ). : 2.1 . : (1) , (i) , +, (ii) , ( ). (2) , (i) , , (ii) , ( ). 2.1 . x (xn ), xn 70

x (, , (xn ) , < x). x (xn ) , (xn ), : , . , , . : (1) 2.1 . (i) : . (ii) : , . : , , ( ) . 2.1 . (2) 2.1 . , , ( ) . : (xn ) : x1 = 1, xn+1 = 2xn .

(xn ) 1, 2 , 2 2 , 2 2 2 , . . . . . , x1 x2 n xn xn+1 . 1 . , , : 2xn 2xn+1 , 2xn 2xn+1 xn+1 xn+2 . xn xn+1 n, , , . xn xn+1 xn 2xn , xn 2 n. (xn ) , , . x (xn ). 2 x2 n+1 = 2xn , x = 2x, x = 0 x = 2. x = 0 1, 1 , , 1. lim xn = 2. 71

: . x1 , x2 , x3 , . . . , {0, 1, . . . , 9}, 9. x1 xn sn = + + n 10 10 x n , xn 0 n, sn+1 = sn + 10n+1 sn n+1 n. (sn ) . , 0 xn 9, 0 sn 1 9 9 9 9 1 10n 1 + + n = 1 < 10 1 = 1. 10 10 10 1 10 1 10

, (sn ) , x, x = lim xn . 0 sn < 1, , 0 x 1. 0x 0 (sn ) n- x. (sn ) , (asn ) . t > x - , t = [x] + 1 - sn < t n, asn < at n (asn ), , . ax . ax = lim asn .1 , a > 1 x < 0, ax = ax . ax a > 1 x. 1.3.. ax ( a > 0 x) ax 1.3..

! : K. ( !) , (1+1)K = 2K. , 1 , (1 + 2 )(1 + 1 )K = (1 + 1 )2 K. 2 2 100 3 , 1 . (1 + 1 )(1 + 3 )(1 + 1 )K = 3 3 1 3 (1 + 3 ) K. , n- n 1 1 (1 + n )n K. , . 1 , xn = (1 + n )n . , K, . , (xn ) , , 73

4. , , .Ac dome giat h (xn ) enai axousa kai nw fragmnh. Gia ton skop aut ja qrhsimopoisoume to exc apotlesma.

Lmma 2.1 Gia kje fusik n kai kje x 1 isqei (1 + x)n 1 + nx.To Lmma 2.1 apodeiknetai pol ekola me thn arq thc epagwgc wc proc ton n.1 n 1 ) (1 + n+1 )n+1 isoduname me thn ( n+1 )n ( n+2 )n+1 ki n n n+1 n+1 n+1 n+2 n+1 n2 +2n n n ( n+1 ) ki aut me thn n+1 ( n2 +2n+1 )n+1 ki aut me thn aut me thn n+1 ( n ) n 1 (1 n2 +2n+1 )n+1 h opoa enai mesh sunpeia tou Lmmatoc 2.1. Prgmati, enai n+1 n+1 1 1 n n+1 1 (1 n2 +2n+1 ) = 1 n+1 = n+1 . 'Ara h akolouja enai axousa. n2 +2n+1 n n 1 Ja apodexoume tra ti (1 + n )n < 4 , to opoo isoduname me 1 < n+1 . Ap 2 n n n to Lmma 2.1 enai n+1 = 1 n+1 n 1 n n+1 n > 1 n n+1 n = n n+1 n+1 n n 1 n( n + 1 n) = 1 n+1+n > 1 2n = 1 . 2

Tra h anisthta (1 +

1 e (1 + n )n .

e = lim 1 +

1 n

n

.

e, , . 1 (1 + n )n , e. e 2, 71828182845904523536 . . . .1 , : 74- (1+ 74 )74 2, 700139678 . . . e. , K , 2, 72K. e , , . , a0 + a1 x + + aN xN = 0 . : m ( m, n) , n , m+(n)x = 0 . . , 2 (2) + x2 = 0. . , e .

74

e y > 0 log y ln y

loge y. , , 1.15 1.16. 2.16 (1) log(yz) = log y + log z y, z > 0. (2) log y = log y log z y, z > 0. z (3) log(y z ) = z log y y > 0 z. (4) log 1 = 0 log e = 1. (5) 0 < y < z, log y < log z. 2.17 a > 0, a = 1. loga y = y > 0. : . 1. , 2, . , 1, P4 , P8 , P16 , . . . 4, 8, 16, . . . . , P2n 2n . P2n P2n+1 : P2n+1 2n P2n 2n P2n , 2n + 2n = 2n+1 . pn P2n , p2 = 4 2. pn+1 pn . 2pn pn+1 = 2+ 4pn 2 4n

log y log a

, , . , , Q4 , Q8 , Q16 , . . . 4, 8, 16, . . . . , Q2n 2n , P2n . qn 75

Q2n , , , q2 = 8 qn pn qn = pn 1pn 2 4n+1

.

, qn+1 = 2+ 4qn 4+qn 2 4n

.

(pn ) (qn ) . , 2pn 4qn 4qn pn+1 = > 2pn = pn qn+1 = < 2+4 = qn . q 22+ 4 pn 4n2

2+ 4

2+

4+

n 4n

qn = 2n+1 pn4

n+1

pn 2

>

2n+1 pn 4n+1

= pn . ,

, : p2 < p3 < < pn < pn+1 < < qn+1 < qn < < q3 < q2 . (pn ) , , ( q2 , ) . 2n . , , 2n . , n , , , . , , , lim pn = 2. lim qn = limpn 1pn 2 4n+1

=

2 14 2 0

= 2.

(pn ) (qn ) lim pn = lim qn = 2 p2 < p3 < < pn < < 2 < < qn < < q3 < q2 . 76

.

Askseic. e. 1. . lim 1 + 1 nn+3

, 1 n

lim 1 +n

1 n+2 2 n

n

, ,1 n

lim 1 + lim 1 =1 n n1 n

1 n+2 2 n1 n

3n+5

,

lim 1

,

lim 1 +

n

. =1 1 1+ n1

(: 1 1 +2 n

=

n n1

=

n+2 n

=

n+1 n+2 n n+1

= 1+

1+

1 n+1

.)

.

1 1. x1 = 1 xn+1 = xn + xn 2 n. (xn ) .

(: . .) 2. 7xn+1 = xn 3 + 6 n. , x1 , (xn ) . (: (xn ), . - x1 , . .) 3. 4xn+1 = xn 2 + 3 n. , x1 , (xn ) . 4. 0 < x1 < 1 xn+1 = 1 1 xn n. (xn ) . 5. > 0, x1 > 0 xn+1 = + xn n. , x1 , (xn ) .n 6. x1 > 0 xn+1 = 6+6xn n. , 7+x x1 , (xn ) .

7. a, x1 > 0 xn+1 = 1 xn + xa n. 2 n (xn ) , , . 77

a ( 1.1 n = 2).1 1 8. x1 = x2 = 1 xn+2 = xn+1 + x1 n. (xn ) n .

78

Keflaio 3

Sunartseic - - . - . . . , . , , . , - - : , ( ), , , .

3.1 Fusik kai gewmetrik paradegmata. : , , p V p V = c, c . , Boyle p V . , p V , , V p : V = c , p 79 p= c . V

V p p V . p , ( ) , V , . . : o C, l l = (1 + )l0 , l0 0o C , , . l , l l . l = 1 l 1 . l0

: , = 2 + 2 2 cos .

. 0o 180o | | + .Epeid enai 1 cos A 1, sunepgetai b2 + g2 2bg cos A b2 + g2 + 2bg = (b + g)2 kai b2 + g2 2bg cos A b2 + g2 2bg = (b g)2 . Epomnwc, |b g| a b + g.

= arccos 2 + 2 2 . 2

, . 0o 180o , , . , = 2 + 2 2 cos cos | | + cos = 2 + 2 2 , 2 80

Ap th dipl anisthta |b g| a b + g sunepgetai b2 + g2 2bg a2 b2 + g2 + 2bg 2 2 2 kai ap autn h 1 b +g a 1. 'Ara uprqei monadik tim tou A ste na enai 2bg b2 +g2 a2 . 0o A 180o kai cos A = 2bg

: 0o 180o ( ) ( ) 180o 0o 360o . : ( ) . , s r s = 2rEdame sto prohgomeno keflaio ti to mkoc enc kklou enai to rio twn mhkn twn eggegrammnwn s' autn kanonikn polugnwn me 2n pleurc. Edame, epshc, ti an sumbolsoume pn aut ta mkh twn polugnwn gia kklo aktnac 1, tte to ri touc, dhlad to mkoc kklou aktnac 1, isotai me 2 diti me to grmma sumbolzoume to mis mkoc tou kklou. An tra jewrsoume opoiondpote kklo aktnac r kai ta antstoiqa eggegrammna s' autn kanonik polgwna me 2n pleurc, tte lgw omoithtac ta antstoiqa mkh autn twn polugnwn enai sa me rpn . 'Ara to mkoc tou kklou aktnac r enai so me lim(rpn ) = r lim pn = r2 .

r= 1 s. 2

s r, r s, , , r s, s r. : A r , A r A = r2 . V r , V r V = 4 3 r . 3

: : t ( k ) 81

q . q t : q = ekt q0 , q0 0. , , q ( ) t ( ). t= 1 q0 log , k q

q t.

3.2 H genik nnoia thc sunrthshc. x ( , x, ) ( ) x y ( , y, ), y x y = f (x) y = F (x) y = g(x)

. f ( ) , , f ( ) y = f (x). x y . y = f (x) ( ) , , ( ) y x. , . : y = x3 + sin(e2x + log2 (x + 3)) y x, f (x) x3 + sin(e2x + log2 (x + 3)), f y = f (x) 82

(, , y = x3 + sin(e2x + log2 (x + 3)) ). , , y = f (x) y = f (x). x y . , , . , , . : u = f (v), t = f (x), y = f (t) . . , , . : . , . ( ) ( ) ( ). . . : .c : V = p . p , ( ) (0, +). , . c , , V = p c , , y = x , , 0 (p x), (, 0) (0, +).

, ( ), . c , y = x , 83

x ( ) x, (, 0) (0, +). . y = f (x) . , . b ( ) a f (a) = b. , y f (x) = y x . : y = 3x 1. (, +) . 3x 1 = y x y . y x = y+1 . y 3 , (, +). , . , . , . : y = 3x 1. (, +) [2, 5). 3x 1 = y x y [2, 5). x = y+1 3 y [2, 5), 2 y+1 < 5 , , 3 7 y < 14. y [7, 14) [2, 5) , , [2, 5) [7, 14).x : y = x1 (, 1) (1, +). x x1 = y x x y . x1 = y (y 1)x = y, , y = 1, , y y = 1, x = y1 . , , y , y1 = 1. , , y = y 1. y = 1 , , (, 1) (1, +).

84

, , (, 1) (1, +) . x (1, +) x1 = y x y (1, +). y , y = 1, x = y1 y (1, +), y1 > 1. 1 y1 > 0 y > 1. y > 1 (1, +) , , (1, +) (1, +). (, 1) (, 1). : y = e2x (, +). y e2x = y x . y 0 e2x = y y > 0 x = 1 log y. y > 0 2 , (0, +). : y = 1+1 x . 1 x 0 , , x

1 x > 0.

x 1 + (, 1] (0, +). y 1+ . , y 0, 1+1 x

1 x

= y x 1+1 x

= y , y < 0.

= y (y 2 1)x = 1. y = 1,

, y 0 y = 1, x = y21 . 1 , y21 1 1 y21 > 0. , y 0, y = 1 1 . , [0, 1) (1, +). (0, +)1 . y 1 + x = y x (0, +). , y < 0 y = 1, x = y21 1 (0, +), y21 > 0. y 2 > 1, 1 y > 1. (0, +) (1, +). 1 , y 1 + x = y x (, 1], (, 1] [0, 1).

85

Askseic.1. . y= 2 x4, 3 y = x2 4x+3, y= 2x 1 , x+4 y= y= x2 1 , x2 + 1 ex + 1 ex 1 y= x2 + 1 , x2 1 x . x1

y = 2x ,

y = log10 x+4,

y = e2x 2ex +3,

y=

2. . (i) y = x2 4x + 3 (, 1], (1, 3], (3, +), (, 2], [2, +). (ii) y = (iii) y =2x1 x+4 x +1 x2 12

(, 4), (4, +). (, 1), (1, 1), (1, +). (, 0), (0, +). [0, 1), (1, +).

ex +1 (iv) y = ex 1 x (v) y = x1

3.3 Analutik kfrash miac sunrthshc. y x . , , : y = x2 , y = sin x, xy = 2, y 2 x3 = 0.

y x : y x . y x: xy = 2 y y y x. : y x 2 y= . x y x. , . y 2 x3 = 0 y, . x > 0 y : 3 3 y = x3 = x 2 y = x3 = x 2 . y 2 x3 = 0 , y = x2 ,3

y = x 2 .

3

, . 86

: . : . , , y 2 x3 = 0 ( ) , , 3 y = x 2 .

Askseic.1. . ( ) x y; ( ;) ; . x + y = 1, x2 2yx + 1 = 0, e(x1)y = x,2

yx = 2, y+x

x3 + y 3 = 0,

x2 y 2 = 0,

(xy)2 = 1,

y 4 2xy 2 + x2 = 1,

sin(x + y) = 1.

3.4 Grafik parstash miac sunrthshc. y = f (x), , . , , 0 . x y = f (x) y ( x y ). x- y-. , x y = f (x) (x, y) = (x, f (x)). , x y = f (x), (x, y) = (x, f (x)) . (x, y) = (x, f (x)) . f = {(x, f (x)) : x f }. , . x, , 87

. (x1 , f (x1 )), (x2 , f (x2 )), . . . , (xn , f (xn )) n x . , .

3.1: . y = f (x), . y = f (x). (x, f (x)) . x- x y- y = f (x) . , y = f (x) x- y-. : y = ax + b a, b . (, +). (0, b). (x, y) y = ax + b , , y b = a(x 0). yb (0, b) (x, y) x0 = a. , (x, y) l 88

(0, b) a. . yb (x, y) l x0 (0, b) yb (x, y) a, x0 = a, y = ax + b , , (x, y) . , , l.

3.2: y = ax + b. : a > 0 a < 0. a > 0, l l , a < 0, l l . a = 0, l l . - , a = 0 - l y- y-, (, +). ( , y = 3x 1 3.2). - , a = 0 - l , y- b, {b}. y = f (x) I , x1 , x2 I x1 < x2 f (x1 ) f (x2 ). x1 < x2 f (x1 ) < f (x2 ), I. , y = f (x) I , x1 , x2 I x1 < x2 f (x1 ) f (x2 ). x1 < x2 f (x1 ) > f (x2 ), I. , . . 89

. , .

3.3: . y y = f (x) x ( ). f (x) = y x y ( ). : y = ax+b (, +), a > 0, , a < 0. a = 0, (, +), x y b. : y = x2 (, +). y = x2 [0, +) (, 0]. [0, +) (, 0] . y x2 = y x [0, +). : y < 0, , y 0, x = y [0, +). [0, +) [0, +). , x2 = y x , y < 0, x = y (, 0], y 0. (, 0] [0, +). 90

3.4: y = x2 .

[0, +) . (0, 0) 1 (0, 0), ( 2 , 1 ), (1, 1), (2, 4). 4 x- [0, +) y- [0, +), [0, +). . , x (x, x2 ) ( y = x2 ) . , (, 0] . , (0, 0) 1 (0, 0), (2, 4), (1, 1), ( 2 , 1 ). x4 (, 0] y- (, 0], [0, +). . , x ( ) (x, x2 ) ( y = x2 ) . y = x2 () . . y = f (x) , f (x) = f (x) x . (a, b) 91

, b = f (a), b = f (a), (a, b) . (a, b) (a, b) y-. , , , y- . y = f (x) y-.

3.5: . y = x2 , y-.

: y = x3 (, +). (, +). 1 1 1 (2, 8), (1, 1), ( 2 , 8 ), (0, 0), ( 2 , 1 ), (1, 1), 8 (2, 8). y = x3 , x3 = y x. y , x = 3 y, y , , (, +). x- , (, +), y- , (, +). . , x ( ) , (x, x3 ) , 92

3.6: y = x3 .

x , (x, x3 ) . y = x3 . y = f (x) , f (x) = f (x) x . (a, b) y = f (x), b = f (a), b = f (a), (a, b) y = f (x). (a, b) (a, b) (0, 0). , , (0, 0) . y = f (x) (0, 0).

y = x3 , (0, 0). : y= 1 x

(, 0) (0, +). 1 y = x (, 0), (0, +). 93

. 1 x = y x. y 0, 1 (0, +) , y > 0, x = y (0, +). (0, +) (0, +). 1 , x = y x (, 0), 1 y 0, x = y (, 0), y < 0. (, 0) (, 0).

3.7: y =

1 x

.

(0, +) . 1 ( 1 , 2), (1, 1), (2, 2 ). 2 x- (0, +) y- (0, +), (0, +). , x 1 , (x, x ) ( 1 y = x ) , 1 x , (x, x ) . , y- x-. , (, 0) . 1 1 (2, 2 ), (1, 1), ( 2 , 2). x- (, 0) y- (, 0), (, 0). , x 94

1 , (x, x ) , x , 1 (x, x ) . , x- y-. 1 y = x , , . 1 y = x , (0, 0): (0, 0). . (a, b) 1 , b = a , a = 1 , (b, a) b . (a, b) (b, a) y = x, . , , . , , .

. , . : 1 y = x . 1 y = x : (, 0) (0, +). 0 . , . , . : y = [x] x, (, +) . y = [x] [k, k + 1), k : y = k x [k, k + 1). [k, k + 1) ( ). , y = [x] [k, k + 1), . , y = [x] ( (, +) ) 95

3.8: y = [x].

.

. . . , , . y = f (x) , , . y = f (x) , , . . , x- y- . . . , [0, +). y- (0, 0). . y = f (x). 96

y = f (x). (1) (x, f (x)) y = f (x) x- (x, f (x)) y = f (x). , y = f (x) y = f (x). x-

3.9: y = f (x) y = f (x).

(2) (x, f (x)) = (x , f (x )) y = f (x) y- (x, f (x)) y = f (x). y = f (x) y = f (x). y-

3.10: y = f (x) + y = f (x ).

(3) . (a, b) (a, b + ). 97

(x, f (x)+) y = f (x)+ (x, f (x)) y = f (x). , y = f (x) + y = f (x).

(4) . (a, b) (a + , b). (x + , f (x)) = (x , f (x )) y = f (x ) (x, f (x)) y = f (x). , y = f (x ) y = f (x).

(5) . (a, b) (a, b). (x, f (x)) y = f (x) (x, f (x)) y = f (x). , y = f (x) y = f (x).

3.11: y = f (x) y = f ( x ).

(6) . (a, b) (a, b).

(x, f (x)) = x , f x y = f x (x, f (x)) y = f (x). , 98

y = f ( x ) y = f (x).

Askseic.1. . y = |x|, y = x [x], y= |x| , x y= 1, x > 0, 0, x = 0, 1 , x < 0, y = (1)[ x ] .1

y = (1)[x] ,

; ; ; . ; 2. y = x2 y = x2 1 . 3. y = x2 , : y = 3x2 , y = x2 4, y = (x+4)2 , y = (3x+4)2 , y = (3x+4)2 4.

, , . 4. a = 0 b, c. y = x2 , y = ax2 + bx + c. (: ax2 + bx + c = a(x + 5. y = y= 1 + 2, x y= 1 , x+21 x b 2 2a )

+

4acb2 4a

.) 3 . x+21 x

, : y= 1 , 3x + 2 y=

6. a, b, c, d c = 0. y = y= (: y=ax+b cx+d ax+b cx+d

,

ax + b . cx + d =bcad 1 c2 x+ d c

=

a ad c (cx+d)+b c

cx+d

+

a c

.)

.

2x+3 y = 3x1 . , , .

99

7. > 0. y = f (x) , y = f (x), y = f (x), y = f (x) + , y = f (x ), y = f (x) y = f ( x ); 8. y = f (x) y = |f (x)|. y = f (x) y = f (|x|). 9. y= 1 , x = , 0 , x = .

3.5 Antstrofh sunrthsh. y = f (x) , , , y x , , x y. , , y y = f (x) x . , y f (x) = y x . , f , f 1 x = f 1 (y) . : y = f (x) x = f 1 (y)

x y f . . f (x) = y x. y x ( y = f (x1 ) = f (x2 ) x1 = x2 ), -- . -- , , . (a, b) y = f (x), b = f (a), a = f 1 (b), (b, a) x = f 1 (y). (b, a) (a, b), y = f (x) x = f 1 (y). , (c, d) x = f 1 (y), d = f 1 (c). c = f (d) , 100

, (d, c) y = f (x). (c, d) , , (d, c), x = f 1 (y) f . , x = f 1 (y) y = f (x). : .

3.12: x = f 1 (y).

: x = f 1 (y) y = f (x), . , x = f 1 (y) y- x-. : , x = f 1 (y) y = f (x), x- y- . x y, y = f 1 (x), x = f 1 (y) y = f (x). , , : x- y- . : 101

, , . , y = f (x) y1 , y2 x = f 1 (y) y1 < y2 . x1 = f 1 (y1 ), x2 = f 1 (y2 ) x = f 1 (y) f (x1 ) = y1 , f (x2 ) = y2 . x1 = x2 , , , y1 = f (x1 ) = f (x2 ) = y2 . x1 > x2 , , y = f (x) , y1 = f (x1 ) > f (x2 ) = y2 . , x1 < x2 , , f 1 (y1 ) < f 1 (y2 ). y = f (x) x = f 1 (y). y = f (x) , x = f 1 (y), , . : , . , y = f (x) , x = f 1 (y), , .

3.13: x =

3

y.

102

: y = x3 . y = x3 , (, +), (, +). , 1 x = 3 y = y 3 , (, +) (, +). . , . , , 1 y = x 3 . y = f (x) --, . , I y = f (x) , x I, y = f (x) --: x1 , x2 I f (x1 ) = f (x2 ) x1 = x2 . y = f (x) I ( ) ( ) I. y = f (x) -- , , . , , I . : y = x2 (, +).