-
, - .
i
-
ii
-
.
. () , , , . , , . .
/ . , , , , . .
, , , . , , Bolzano - Weierstrass , , .
; - . , , , - , , . , , , . .
.
1. , , - - : - , , . . , , . , Supremum: - .2. . , n n 2.2, 2.3.
, 3.5, 3.6.3. , , 10 . , , - .
iii
-
4. Riemann , , Darboux . , Riemann . Riemann , Darboux - Darboux
Riemann.5. . , -. , (, ), .6. , , ). , ( , Jensen, , Riemann , -, )
. : . .6. Peano. , , . Foundations of Analysis E.Landau . () ()
Dedekind. , , Cauchy , ., , : , .7. . , Cauchy .8, , , , , . : - ,
, - .
, , :Mathematical Analysis, T. Apostol.Differential and Integral
Calculus, R. Courant.The Theory of Functions of Real Variables, L.
Graves.
iv
-
Foundations of Analysis, E. Landau.Principles of Mathematical
Analysis, W. Rudin.A Course of Higher Mathematics, V.
Smirnov.Calculus, M. Spivak.The Theory of Functions, E. C.
Titchmarsh.
, , Calculus, T. Apostol.Introduction to Calculus and Analysis,
R. Courant - F. John., , , . (http://users.uoa.gr/apgiannop/). , .
- . - . .
, , ., , , .
2011.
v
-
vi
-
I : . 1
1 . 31.1 R . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 31.2 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 51.3 . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 101.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . .
. 121.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 181.5 Supremum infimum. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 20
2 . 272.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 27
2.1.1 , , . . . . . . . . . . . 272.1.2 n N . n N. . . . . .
29
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 322.3 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 372.4 . . . . . . . . . . . . . . .
. . . . . . . . . . 382.5 . e, . . . . . . . . . . . . . . . . . .
. . . . . . . 512.6 Supremum, infimum . . . . . . . . . . . . . . .
. . . . . . . . . . . . 602.7 . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 612.8 . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 672.9 limsup liminf . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 69
3 . 773.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
773.1.2 . . . . . . . . . . . . . 82
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 883.2.1 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 883.2.2 . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 923.2.3 , , . . . . . . . . . . . .
. . . . . . . . . 94
3.3 . . . . . . . . . . . . . . . . . . . . . . . . 963.4 . . .
. . . . . . . . . . . . . . . . . . . . . . . . 1073.5 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 1093.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 111
vii
-
3.5.3 . . . . . . . . . . . . . . . . . . . . . 1123.5.4 . . . .
. . . . . . . . . . . . . . . . . . . . . . 114
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1163.7 Cauchy. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 119
4 . 1214.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 121
4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1214.1.2 . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 123
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1284.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1364.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.6 .
. . . . . . . . . . . . . . . . . . . . . . . . . . 151
II : . 157
5 . 1595.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 159
5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 1595.1.2 . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1615.1.3 , . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 164
5.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1675.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 1695.4 , . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 1765.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1785.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 184
5.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1845.6.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 186
5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 1905.7.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1925.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . 1925.7.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1975.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1975.7.5 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 200
5.8 . . . . . . . . . . . . . . . . . . . . . . . . . 2045.8.1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2045.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 209
5.9 , . . . . . . . . . . . . . . . . . . . . . . . . . 2125.9.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2125.9.2 . . . . . . . . . . . 215
5.10 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 218
6 Riemann. 2236.1 Darboux. . . . . . . . . . . . . . . . . . . .
. . . . . . 2236.2 . Darboux. . . . . . . . . . . . . . . . . . . .
. . . . . 228
6.2.1 . . . . . . . . . . . . . . . . . . . . . . 228
viii
-
6.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 2326.3 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 2356.4 . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 2376.5 . Riemann. . . . . . . . . . . . . . . .
. . . . . . . . . 254
7 . 2617.1 , . . . . . . . . . . . . . . . . . . . . . . . .
261
7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 2617.1.2 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 262
7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 2677.3 . . . . . . . . . . . . . . . . . . . . . . . . .
272
7.3.1 . . . . . . . . . . . . . . . 2727.3.2 . . . . . . . . . .
. 2747.3.3 . . . . . . . . . . . . . . . . . . . . . . 2757.3.4 . .
. . . . . . . . . . . . . . 2807.3.5 . . . . . . . . . . . . . .
285
7.4 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 2927.5 . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 293
7.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
2937.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . 2957.5.3
. . . . . . . . . . . . . . . . . . . 2967.5.4 Jensen. . . . . . .
. . . . . . . . . . . . . . . . . . . . . 298
III : . 299
8 . 3018.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 3018.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 3068.3 p- . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 3118.4 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 316
8.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 3178.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 3188.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 319
8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 3248.5.1 . . . . . . . . . . . . . . . . . . . . . . .
3248.5.2 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 3298.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 331
9 . 3379.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 3379.2 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 3399.3 Weierstrass. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 351
10 . 35510.1 . . . . . . . . . . . . . . . . . . . . . . .
35510.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 36110.3 Taylor. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 37410.4 . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 381
ix
-
10.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
38110.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 384
11 . 38711.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 38711.2 - . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39511.3 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 397
11.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 39811.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 399
11.4 . . . . . . . . . . . . . . . . . . . . . . . 40211.4.1 . .
. . . . . . . . . . . . . . . . . . . 40211.4.2 . . . . . . . . . .
. . . . . . . . . . . . . . . 40511.4.3 . . . . . . . . . . . . . .
. . . . . 407
11.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 414
IV . 417
12 . 41912.1 Peano. . . . . . . . . . . . . . . . . . . . . . .
. 419
12.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41912.1.2 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 42212.1.3 . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 423
12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 42512.2.1 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 42512.2.2 . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 42612.2.3 . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 42812.2.4 . . . .
. . . . . . . . . . . . . . . . . . . 430
12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 43112.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 43212.3.2 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 43312.3.3 . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 43512.3.4 R+ . . . . . .
. . . . . . . . . . . . . . . . . . . 43712.3.5 . . . . . . . . . .
. . . . . . . . . . . . . . . 438
12.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 44112.4.1 . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 44112.4.2 . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 44212.4.3 . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 44312.4.4 . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 44412.4.5 R. .
. . . . . . . . . . . . . . . . . . . . . . . . 444
12.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 44512.5.1 Cauchy. . . . . . . . . . . . . . . . . . . . . .
44512.5.2 . . . . . . . . . . . . . . . . . . 446
12.6 : R N. . . . . . . . . . . . . . . . . . . . . 44712.6.1 R.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44712.6.2 R. . . . . . . . . . . . . . . . . . . . . . . . . .
44812.6.3 , , . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 451
x
-
I
: .
1
-
2
-
1
.
1.1 R . R. ,
, , N, Z Q. R, .
R , . .
. R = R {,+}
R. +, ,.
R. , . ( ) ( ) . , : (+) + (+) = +. , , : (+)(+) . , ( ) , , :
(+)0 , , . .
. , + , +. :
< x, x < +, < +.
(+) = , () = + .
3
-
(+) + x = +, x+ (+) = +, (+) + (+) = +,
() + x = , x+ () = , () + () = .
, (+) + (), () + (+)
.
(+) x = +, x () = +, (+) () = +,
() x = , x (+) = , () (+) = .
(+) (+), () ()
.
()x = , x() = (x > 0),
()x = , x() = (x < 0),
()() = + , ()() = .
()0, 0()
.
1
+= 0,
1
= 0.
1
0
. .
x
= (x > 0), x
= (x < 0), x
= 0.
x
0,
0
,
,
. x0= x1
0
10.
0= ()1
0 1
0. = ()
1 = ()0
= ()1 = ()0 .
,
|+| = +, | | = +.
4
-
. 1. , , - (a, u, x,m, n, , ) (M,S ) , R. , , , , R. , . : a R,
(a,+], A R, A [, 3].2. N = {1, 2, 3, . . . }. , 0 .
.
1. - ( ) ( ) ( ) .
2. R R . , (, , ) R R , : . : (xy)z =x(yz) x, y, z R , xy, yz,
(xy)z, x(yz).
3. x y < 0 z w < 0, 0 < yw xz.
4. (i) x y, z w, t s x+ z+ t = y+w+ s, x = y, z = w t = s.(ii) 0
< x y, 0 < z w, 0 < t s xzt = yws, x = y, z = w t = s.
5. (i) |x| a a x a.(ii) |x| < a a < x < a.(iii) :
|x| |y| |x y| |x|+ |y|.(iv) |x+ y| = |x|+ |y| x, y 0 x, y 0.(v)
|x+y+z| |x|+ |y|+ |z|. , |x+y+z| = |x|+ |y|+ |z| x, y, z 0 x, y, z
0.
6. , a x b a y b, |x y| b a.
7. (i) x : |x+1| > 2, |x 1| < |x+1|,xx+2
> x+33x+1
, (x 2)2 4, |x2 7x| > x2 7x, (x1)(x+4)(x7)(x+5) > 0,
(x1)(x3)(x2)2 0.
(ii) x x : (, 3], (2,+),(3, 7), (,2) (1, 4) (7,+), [2, 4] [6,+),
[1, 4) (4, 8], (,2] [1, 4) [7,+).
1.2 . R, .
, , :
5
-
, , . , A,B - A B, : , .
. - A,B a b a A, b B. a b a A, b B.
R . , 1.5 Q .
1.1 .
1.1. b n N n > b.
. - - b n b n N.
B = {b |n b n N}
. , n b n N, b B. ,
n b (n N, b B).
1 < , 1 B. n N n > 1 , ,n+ 1 > . , n+ 1 N.
, , 0.
. a > 0 n N 1n< a .
. 1.1 b = 1a.
1.1 .
1.1. x k Z k x < k + 1.
. 1.1 n N n > x m N m > x. l = m,
l < x < n.
l, n . , k Z, k x k + 1 x. l x, k x k Z, k l. , , n > l n
> x. , , k Z k x k + 1 > x,
k x < k + 1.
k k x < k + 1 .
k x < k + 1, k x < k + 1
6
-
k, k Z. k < k + 1, k < k + 1,
1 < k k < 1.
k k Z, k k = 0,
k = k.
. k Z k x < k + 1, 1.1, x [x].
, [x] x < [x] + 1, [x] Z.
Q . , , .
. a, b, a < b r Q a < r < b.
. 1.1, n N
n > 1ba .
m = [na] + 1.
na < [na] + 1 = m na+ 1 < nb
, , a < mn< b. r = m
n a < r < b.
, , . -, , , : () . .
.
1. (i) a > 0, a 0.: a > 0. > 0 < a.(ii) a b+ > 0,
a b.(iii) |a b| > 0, a = b.
2. [a, b) (a,+) .
3. A = (, 0], B = [0,+) : a b a A, b B. A = (, 0], B = (0,+), A
= (4,2), B = (2,+) A = (, 0), B = [1, 13].
7
-
4. - A,B AB = , AB = R a b a A, b B. A = (, ), B = [,+) A = (,
], B = (,+).
5. (i) - A,B a b a A, b B > 0 a A, b B b a . a b a A, b
B.(ii) - A,B 0 < a b a A, b B > 0 a A, b B b
a 1 + .
a b a A, b B.
6. (i) a 1n n N, a 0.
(ii) a b+ 1n n N, a b.
(iii) |a b| 1n n N, a = b.
1: 1 R .
7. A = { 1n|n N}, B = { 1
n|n N}
a b a A, b B. A = {r Q | r < 0}, B = {r Q | r > 0}.
8. (i) x a x < b, b a.(ii) r a r Q, r < b, b a.(iii) {r Q
| r > a} = {r Q | r > b}, a = b.(iv) {r Q | r < a} {r Q |
r > b} = , a b.(v) {r Q | r a} {r Q | r b} = Q, b a.
9. (i) , , . 6= 0 .(ii) , , .
10. (i) [b] n Z n > b.(ii) [b] n Z n b.(iii) [b] n N n >
b.(iv) [b] n N n b.(v) a > 0, [ 1
a] n N 1
n< a.
11. k Z, [x+ k] = [x] + k.
12. [x+ y] = [x] + [y] [x+ y] = [x] + [y] + 1. [x+ y + z].
13. [x] + [x+ 1n] + + [x+ n2
n] + [x+ n1
n] = [nx] n N, n 2.
14. k N, a < b. ba, r (a, b) r = m
n, m,n Z, 1 n k.
8
-
1.3 .1.3.1 .
. , n N, an a n :an = a a (n N),
n a., a 6= 0, n Z, n < 0, a0 , an :
a0 = 1, an =1
an=
1
a a(a 6= 0, n Z, n < 0),
|n| = n a.
00
., n Z , (a)n = an , n Z ,
(a)n = an . , n Z , an > 0 a 6= 0. n Z , an > 0, a > 0,
an < 0, a < 0.
1.2 . ! 1.2. (1) :axbx = (ab)x , axay = ax+y , (ax)y = (ay)x =
axy .(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.
. (1) : x > 0,
axbx = (a a)(b b) = (ab) (ab) = (ab)x . x > 0. : x, y >
0,
axay = (a a)(a a) = a a = ax+y . x, y > 0. : x, y > 0,
(ax)y = ax ax = (a a) (a a) = a a = axy . x, y > 0 (ay)x =
axy (ax)y = axy x, y.(2) (i) a < b x ,
ax = a a < b b = bx . (iii) (i) (ii) .(3) (i) 1 < a y x
,
ax = (a a)(1 1) < (a a)(a a) = ay . (iii) (i) (ii) .
. - , 1.2, , R . , , 1.2 .
9
-
1.3.2 .
, 1 6 1.2. .
Bernoulli. a 1,
(1 + a)n 1 + na
n N. a > 1, a 6= 0 a = 1, n 2, .
. .
1.2 , n , a 0 xn = a - .
1.2. n N, a 0 x 0 xn = a.
. a = 0, , , xn = a 0. a > 0. 0 n = a, , , > 0.
Y = {y | y > 0, yn a}, Z = {z | z > 0, zn a}.
y0 = min{a, 1}, z0 = max{a, 1}.
y0 Y z0 Z , Y, Z . y Y ,z Z yn a zn, yn zn , y, z > 0, y z.
,
y z (y Y, z Z).
n = a. 0 < < . < , Z. 0 ( )n < a. ,
( )n < a. Bernoulli
an>
(1
)n 1 n ,
nann1
< .
> 0, < , > 0.
nann1
0,
n a.
> 0. + > , + Y . + 0 ( + )n > a. ,
( + )n > a.
10
-
Bernoulli
n
a>
(+
)n=
(1
+
)n 1 n +
,
anna
< +
<
, ,(an)na
< .
> 0, (an)na
0, ,
n a. n a n a
n = a.
xn = a. 1, 2 > 0, 1n = a 2n = a, 1n = 2n , 1 = 2 .
1.2 4.4.
. n N, a 0, - xn = a, 1.2, n- a
na (a 0, n N).
, n0 = 0 n
a > 0, a > 0.
1.2 - xn = a. 1.3 ; , 1.2, .
1.3. n N , xn = a , na
na , a > 0, , n
0 = 0, a = 0, , a < 0.
n N , xn = a , na , a > 0,
n0 = 0, a = 0, n
a , a < 0.
- - . .
1.4. n, k. nk k n-
.
. k n- , m N
k = mn .
nk = m
, , ., n
k .
nk = m
l,
11
-
m, l N - - m, l > 1. - - l > 1, p l. l,m > 1, p m. k =
mn
ln,
lnk = mn .
p l, lnk , , mn . , , . p mn = m m, m . l = 1,
k = rn = mn
n- .
, , R . 1.2 , , .
1.5. R \Q .
. 2 , 1.4 2
.
, , , Q : 2. 1.2 Q: , x2 = 2 Q.
, , , : , , .
. a, b, a < b x R \Q a < x < b.
. a+2 < b+
2, r Q a+
2 < r < b+
2 . (r
2) R\Q
a < r 2 < b.
1.3.3 .
1.1. a > 0, m, k Z, n, l N mn= k
l. ( n
a)m = ( l
a)k .
.
(( na)m)nl = ( n
a)mnl = (( n
a)n)ml = aml , (( l
a)k)nl = ( l
a)knl = (( l
a)l)kn = akn .
ml = kn, aml = akn , (( na)m)nl = (( l
a)k)nl . , ( n
a)m > 0
( la)k > 0, ( n
a)m = ( l
a)k .
. , a > 0, r Q. m Z, n N r = mn.
, 1.1, ( na)m . ,
ar = ( na)m (a > 0, r = m
n,m Z, n N).
, r Q, r > 0,
0r = 0 (r Q, r > 0).
12
-
, 0r , 0r = ( n0)m r = m
n, m,n N. ,
n N a 1n = na (a 0, n N). , ar ,
ar > 0 a > 0, r Q.
1.2 . . x = m
n y = k
l, m, k Z, n, l N.
(ax)n = am :
(ax)n = (( na)m)n = (( n
a)n)m = am .
(1) : a, b > 0.
(axbx)n = (ax)n(bx)n = ambm = (ab)m = ((ab)x)n
, axbx > 0 (ab)x > 0, axbx = (ab)x . a = 0 b = 0 . : x+ y
= ml+kn
nl. a > 0.
(axay)nl = (ax)nl(ay)nl = ((ax)n)l((ay)l)n = (am)l(ak)n = amlakn
= aml+kn = (ax+y)nl
, axay > 0 ax+y > 0, axay = ax+y . a = 0 . : xy = mk
nl. a > 0.
((ax)y)nl = (((ax)y)l)n = ((ax)k)n = ((ax)n)k = (am)k = amk =
(axy)nl
, (ax)y > 0 axy > 0, (ax)y = axy . (ay)x = axy (ax)y = axy
x, y. a = 0 .(2) (i) x > 0, m > 0.
(ax)n = am < bm = (bx)n
, ax > 0 bx > 0, ax < bx . (iii) (i) (ii) .(3) (i)
(ax)nl = ((ax)n)l = (am)l = aml , (ay)nl = ((ay)l)n = (ak)n =
akn .
x < y n, l > 0, ml < kn. aml < akn , (ax)nl <
(ay)nl . ax > 0 ay > 0, ax < ay . (iii) (i) (ii) .
1.3.4 .
a > 1. r, s, t Q, s < r < t as < ar < at . , s, t
Q, x R \ Q, s < x < t as < ax < at . as , at ax . , ax
: as < ax < at s, t Q, s < x < t.
1.2. a > 1.(1) b > 1 n N bn > a.(2) b a 1n n N, b
1.
13
-
. (1) n N n > a1
b1 .
, Bernoulli,
bn = (1 + b 1)n 1 + n(b 1) > a.
(2) (1).
1.3. (1) a > 1, x R \ Q. as < < at s, t Q, s < x
< t.(2) a > 1, x Q. as < < at s, t Q, s < x < t
ax .
. (1) x R \Q.
S = {as | s Q, s < x}, T = {at | t Q, t > x}.
S, T , s, t Q s < x < t. , S T . , s < x < t s <
t , a > 1, s, t Q, as < at . S, T ,
as at (s, t Q, s < x < t).
as < < at . , , s, t Q
s < s < x < t < t,
as < as
at < at
, ,as < < at (s, t Q, s < x < t).
(2) x Q. = ax , as < < at s, t Q, s < x < t. , (1)
(2), . 1, 2 as < 1 < at as < 2 < at s, t Q, s < x
< t.
21< ats , 1
2< ats
s, t Q, s < x < t. , n N s, t Q
x 12n< s < x < t < x+ 1
2n.
t s < 1n, ,
21< a
1n , 1
2< a
1n .
n N, 1.2 21
1, 12
1.
1 = 2 .
14
-
. a > 1, x R \Q, ax 1.3.
ax = (a > 1, x R \Q).
, , ax as < ax < at s, t Q,s < x < t . 1.3, a >
1, x Q ax .
9 1.5 ax , a > 1 x R \Q.
. a = 1, x R \Q,
1x = 1 (x R \Q).
0 < a < 1, x R \Q, 1a> 1, ( 1
a)x .
ax =1
(1/a)x(0 < a < 1, x R \Q).
, x R \Q, x > 0,
0x = 0 (x R \Q, x > 0).
, , ax a > 0, a = 0, x > 0 a < 0, x Z. , ax a = 0, x 0
a < 0, x R \ Z.
. 4.3 - ax a < 0 x .
ax . a > 0. x Q, ax > 0. x R \ Q, , ax , ax > as s Q, s
< x,, as > 0, ax > 0. ax > 0 a > 0 x.
1.3. (1) s, t Q, s < x + y < t. s, s, t, t Q s < x <
t,s < y < t , s = s + s , t = t + t.(2) x, y > 0, s, t Q,
0 < s < xy < t. s, s, t, t Q 0 < s < x < t,0 <
s < y < t , s = ss , t = tt.
. (1) , s Q
s y < s < x
s = s s Q.
s < x, s = s + s s = s s < y. , t Q
x < t < t y
t = t t Q.
15
-
x < t, t = t + t y < t t = t.(2) s Q
sy< s < x
s = s
s Q.
s < x, s = ss s = ss< y. , t Q
x < t < ty
t = t
t Q.
x < t, t = tt y < tt= t.
1.2 . - .(1) : a, b > 1. s, t Q, s < x < t as < ax
< at bs < bx < bt ,
(ab)s = asbs < axbx < atbt = (ab)t .
axbx (ab)s , (ab)t s, t Q, s < x < t, (ab)x
axbx = (ab)x .
- - a, b > 1. : a > 1. s, t Q, s < x+ y < t , 1.3,
s, s, t, t Q s < x < t, s < y < t , s = s + s , t = t +
t . as < ax < at
as < ay < at , ,
as = asas
< axay < at
at
= at .
axay as , at s, t Q, s < x + y < t, ax+y
axay = ax+y .
a > 1. : a > 1, x, y > 0. s, t Q, s < xy < t s1 Q
s1 s, 0 < s1 < xy < t. 1.3, s, s, t, t Q 0 < s < x
< t, 0 < s < y < t , s1 = ss , t = tt. 1 < as
< ax < at
, ,
as as1 = (as)s < (ax)s < (ax)y < (ax)t < (at)t = at
.
(ax)y as , at s, t Q, s < xy < t, axy
(ax)y = axy .
a > 1, x, y > 0 (ay)x = axy (ax)y = axy x, y.(2) (i) s Q 0
< s < x, , 1 < b
a,
1 < ( ba)s < ( b
a)x .
16
-
,ax < ax( b
a)x = (a b
a)x = bx .
(ii) (iii) (i).(3) (i) r Q x < r < y, ax < ar < ay .
(ii) (iii) (i).
(2) 1.2 ax a (0,+), x > 0, 1 (0,+), x = 0, (0,+), x < 0.
(3) 1.2 ax x (,+), a > 1, 1 (,+), a = 1, (,+), 0 < a < 1.
ax a [0,+), x > 0.
. , ,
a+ = 0 (0 a < 1), a+ = + (a > 1),a = + (0 < a < 1),
a = 0 (a > 1),
(+)b = + (b > 0 b = +), (+)b = 0 (b < 0 b = ),
, 00 ,
1 , (+)0 , 0
.
.
1. Bernoulli 1.3. 1.2, .
2. S = {s Q | s
2}.
S, T Q s t s S , t T . Q s t s S , t T . Q .
3. - ( ) - ( ) ( ) 1 . .
4. (i) a 2 > 0, a 0.(ii) a 1 + + 2 > 0, a 1.
5. 0 < a < 1. ax at < < as s, t Q,s < x <
t.
6. n N (1 + a)n 1 + na + n(n1)2
a2 a 0 (1 + a)n 1 + na+ n(n1)
2a2 + n(n1)(n2)
6a3 a 1. .
17
-
7. n N , xn < yn x < y. n N , xn < yn |x| < |y|.
8. , n N, n 2,
xn yn = (x y)(xn1 + xn2y + + xyn2 + yn1)
, n N, n 3 ,
xn + yn = (x+ y)(xn1 xn2y + xyn2 + yn1).
9. x, y 0, x2+xy+y2 > 0 x4+x3y+x2y2+xy3+y4 >0. ; x3 + x2y
+ xy2 + y3 > 0 x5 + x4y + x3y2 + x2y3 + xy4 + y5 > 0; ;
10. n N n! = 1 2 n 0! = 1.
(nm
)= n!
m!(nm)! m,n Z, 0 m n.(i)
(nm
) n Z, n m (
nm
) m Z, 0 m n.
(ii) (nm
)+(
nm1
)=
(n+1m
) m,n Z, 1 m n.
(iii) Newton: x, y n N
(x+ y)n =
(n
0
)xn +
(n
1
)xn1y + +
(n
n 1
)xyn1 +
(n
n
)yn =
nk=0
(n
k
)xnkyk .
: .(iv) n N
nk=0
(nk
)= 2n ,
nk=0
(nk
)(1)k = 0.
11. , :(i) n
a nb = n
ab n
ma = m
na = nm
a, a 0, n,m N.
(ii) na < n
b , n N, 0 a < b.
12. n N , nan = |a|.
13. na+ b n
a + n
b n N, n 2, a, b 0.
na+ b = n
a+ n
b a = 0 b = 0.
14. 3 , 7
129 ,
2 + 3
5 3
2 +
5 .
15. 105105 106
106 .
1.4 . 1.4 , a > 0, a 6= 1, y > 0 ax = y
.
1.4. a > 0, a 6= 1. y > 0 x ax = y.
18
-
. a > 1, y 1.
U = {u | au y}, V = {v | av y}., 0 U . , 1.2, n N an > y , ,n
V . U, V . u U , v V au y av , au av , a > 1, u v. U, V ,
u v (u U, v V ). a = y. n N. 1
n< , 1
n V .
a1n < y.
, < + 1n, + 1
n U .
y < a+1n .
ya< a
1n , a
y< a
1n
n N , 1.2,ya
1, ay 1.
a = y.
: a > 1, 0 < y < 1, , 1
y> 1, a = 1
y, , =
a = a = y. 0 < a < 1, , 1
a> 1, ( 1
a) = y, =
a = a = y., ax = y . a1 = y, a2 = y, a1 = a2 , 1 = 2 .
, a > 0, a 6= 1, ax loga y .
. y > 0 a > 0, a 6= 1, ax = y, 1.4, y a
loga y.
y 0 a > 0, a 6= 1, ax = y . 9 1.5 loga y. 1.6 .
1.6. a, b > 0, a, b 6= 1.(1) loga(yz) = loga y + loga z y, z
> 0.(2) loga
yz= loga y loga z y, z > 0.
(3) loga(yz) = z loga y y > 0 z.(4) logb y =
loga yloga b
y > 0.(5) loga 1 = 0, loga a = 1.(6) 0 < y < z. (i)
loga y < loga z, a > 1, (ii) loga y > loga z, 0 < a
< 1.
19
-
. (1) x = loga y, w = loga z, ax = y, aw = z. ax+w = axaw = yz,
loga(yz) = x+ w = loga y + loga z.(2) loga
yz+ loga z = loga(
yzz) = loga y, loga
yz= loga y loga z.
(3) x = loga y, ax = y. azx = (ax)z = yz , , loga(yz) = zx =z
loga y.(4) x = logb y, w = loga b, bx = y, aw = b. awx = (aw)x = bx
= y. loga y = wx = loga b logb y.(5) loga 1 = 0 a0 = 1 loga a = 1
a1 = a.(6) x = loga y, w = loga z, y = ax , z = aw . ax < aw , a
> 1, x < w , 0 < a < 1, x > w.
(6) 1.6 loga y (0,+), a > 1, (0,+), 0 < a < 1.
.
1. ax = y a 0, a = 1, y 0;
2. log2 3 log3 5 log5 7 log7 10 log10 8.
3. log2 3 ;
4. a > 0, a 6= 1. log 1ay = loga y y > 0.
5. a > 0, a 6= 1. logaz(yz) = loga y y > 0 z 6= 0.
1.5 Supremum infimum.. - A. A u u a a A , , A (, u]. u A. , A l
l a a A , , A [l,+). l A. , A , l, u A [l, u].
u A, u u , , A , l A, l l A.
. (1) [a, b] u b . (a, b], [a, b), (a, b), (, b], (, b). : ,
b.(2) [a, b], (a, b], [a, b), (a, b), (a,+), [a,+) l a . : , a.(3)
, (a,+), [a,+), (,+) (, b), (, b], (,+) .(4) N , 1 , N l 1 . , 1 N.
, 1.1 , , N :
20
-
u N u n N n > u. , N !!!
1.5. - A.(1) A , A .(2) A , A .
. (1)
U = {u |u A}.
U A. , a u a A, u U . ,
a u (a A, u U).
a a A, A. u u U , A.(2) : U , L = {l | l A}.
. -, A supremum A
supA.
-, A infimum A
infA.
. [a, b], (a, b), (a, b], [a, b) (, b], (, b) supremum, b. , [a,
b], (a, b), (a, b], [a, b), (a,+), [a,+) infimum, a.
. , , A maximum A
maxA.
, , , A minimum A
minA
1.7. (1) maxA, supA = maxA.(2) minA, infA = minA.
. (1) maxA A. A maxA, maxA A.(2) .
21
-
. (1) A = {0} [2, 3] {4} minA = 0 maxA = 4. infA = 0 supA =
4.(2) minN = 1, infN = 1.(3) A = { 1
n|n N} = {1, 1
2, 13, 14, . . . } maxA = 1, supA = 1. A
. infA., l 0 A. l > 0, , , n N 1
n< l, l A.
A - , A 0. , infA = 0.
. - A ,
supA = +.
, - A ,
infA = .
. A , . , + , , , , , A.
. N , supN = +.
. , supA, , +. , , supA = u ( ) supA u, supA = u R . infA.
1.8. - A supremum infimum
infA supA.
. - A , , 1.5, supA . A , , , supA = +. , - A , infA , , A ,
infA = ., a A.
infA a supA, infA supA.
. A , infA, supA , , infA = supA. A , infA < supA.
1.9 , , supremum infimum.
1.9. - A.(1) a supA a A. , u < supA a A a > u ,, u < a
supA.(2) a infA a A. , l > infA a A a < l ,, infA a <
l.
22
-
. (1) A , supA ( ) A, a supA a A. A , supA = + , , a supA a A.,
u < supA. supA A, u A, a A a > u.(2) .
1.5 .
Supremum. -, .
, Supremum . -, 1.10 Supremum. Supremum .
1.10. -, . .
. A,B - a b a A, b B., b B A , B , A . A , supA. supA A,
a supA
a A. b B A supA A,
supA b
b B. , = supA,
a b
a A, b B.
, , Infimum, -, . 1.5 . 1.10, Infimum . 15.
. , - - .
A . A : x1, x2 A, x1 < x2 x x1 < x < x2 x A. : . 1.11 ,
- R, .
1.11. - A : x1, x2 A, x1 < x2 x x1 < x < x2 x A. A
.
23
-
. u = supA, l = infA,
l u +. , ,
A [l, u].
x (l, u). x A, x1, x2 A,
x1 < x < x2 .
, x A. ,
(l, u) A.
(l, u) A [l, u]
: A = (l, u), A = [l, u], A = (l, u] A = [l, u). A .
.
1. 1.5 1.7 1.9.
2. max{x, y} = x+y+|xy|2
min{x, y} = x+y|xy|2
.
3. (i) t x, t y t z t min{x, y, z}.(ii) t x, t y t z t max{x, y,
z}.
4. (a,+), (a, b), (a, b] l a;
5. (i) infima suprema {1, 0, 2, 5}, [1, 5], (1, 5), (1, 0] (2,
5]. -;(ii) infA = infB supA = supB. A = B;
6. infima suprema {(1)nn |n N}, {1+(1)n
2n+ 1(1)
n
2n |n N},
{(1)n + 1n|n N}, {n(1)
n(n1)2n
|n N}, { 1n+ 1
m|n,m N}, { 1
n 1
m|n,m N},
{x + y | 0 < y < 1, 4 < x < 5}, {x y | 0 < y <
1, 4 < x < 5},+n=1[2n 1, 2n],+
n=1[12n, 12n1 ].
7. - A. A, : supA = +, supA < +.
8. a < b A = {r Q | a < r < b}. infA supA.
9. (i) a > 1, sup{as | s Q, s < x} = ax = inf{at | t Q, t
> x}. - - ax x. a = 1 0 < a < 1;(ii) y > 0, a > 1,
sup{u | au y} = loga y = inf{v | av y}. - - loga y. 0 < a <
1;
24
-
10. A = [0, 2], A = [0, 2), A = [0, 1] {2} 1.9, , , . - A u =
supA.(i) A (u , u] 6= > 0;(ii) A (u , u) 6= > 0; , , u / A; l
= infA.
11. - A.(i) supA u a u a A.(ii) u supA < u a A, a > .
infA.
12. - A,B.(i) supA infB a b a A, b B.(ii) a b a A, b B. a b a A,
b B.(iii) a b a A, b B > 0 a A,b B b a . supA = infB. (i) 5
1.2.
13. - A,B, A B. infB infA supA supB.: supB B, A.
14. - A,B. A supA supB a A < a b B, b > .
15. Infimum.
16. - A,B. sup(AB) = max{supA, supB} inf(AB) =min{infA,
infB}.
17. (i) - A A = {a | a A}. sup(A) = infA inf(A) = supA.(ii) -
A,B A + B = {a + b | a A, b B}. sup(A+B) = supA+ supB inf(A+B) =
infA+ infB.(iii) - A,B (0,+) A B = {ab | a A, b B}. inf(A B) = infA
infB sup(A B) = supA supB.
18. ; , inf , sup ; inf sup ;
25
-
26
-
2
.
2.1 .2.1.1 , , .
. ( )
x : N R
N . n N x(n) , , xn . ,
xn = x(n) (n N).
, : x1 , x2 , x3 . , / . xn+1 xn xn1 xn . n, - N, . x : N R
(x1, x2, . . . , xn, . . . ), (xn), (xn)+n=1 .
, , , x, n, : (yn), (xk), (zm) .
. (1) ( 1n) (1, 1
2, 13, . . . , 1
n, . . . ).
(2) (n) (1, 2, 3, 4, . . . , n, . . . ).(3) (1) (1, 1, 1, . . .
, 1, . . . ).(4) ((1)n1) (1,1, 1,1, . . . , 1,1, . . . ).(5)
(1
10n
) ( 1
10, 1102
, 1103
, . . . , 110n
, . . . ).(6) n- n, (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, . . .
).(7) (m n)+n=1 (m 1,m 2,m 3, . . . ,m n, . . . ).(8) (m n)+m=1 (1
n, 2 n, 3 n, . . . ,m n, . . . ). (xn)+n=1 , (xn), - - .
27
-
: . , , . (1)+n=1 {1} . , , (1, 1, 1, . . . ). , . , . , (1,1,
1,1, 1,1, . . . ),(1, 1,1, 1, 1,1, 1, 1,1, . . . ) {1, 1}. : (1,
1
2, 13, 14, 15, 16, . . . ), (1
2, 1, 1
4, 13, 16, 15, . . . )
{1, 12, 13, 14, 15, 16, . . . } = { 1
n|n N}.
. (xn) xn+1 xn n N, xn+1 > xn n N, xn+1 xn n N, xn+1 < xn
n N. (xn) , . (xn) , c xn = c n N.
, . , , . .
. (xn) , u xn u n N. u (xn). (xn) , l xn l n N. l (xn). (xn) ,
l, u l xn u n N.
, u (xn), u u , l (xn), l l .
. (1) (c) .(2) ( 1
n),( (1)n1
n
), (n1
n), ((1)n1)
[1, 1].(3)
( (1+(1)n1)n2
) (1, 0, 3, 0, 5, 0, 7, 0, . . . ) -
. , l 0 . - - u . (1+(1)
n1)n2
u n N,, n = 2k 1, 2k 1 u k N. k u+1
2
k N. .(4) (1, 0,3, 0,5, 0,7, 0, . . . ), , . l , l, l, .(5)
((1)n1n) (1,2, 3,4, 5,6, . . . ) -. , u l , , (1)n1n u n N l (1)n1n
n N. , n N,.
28
-
(xn) , l, u
l xn u
n N. M = max{u,l},
M u M l , , M xn M , ,
|xn| M
n N. , (xn) , M |xn| M n N. : M
|xn| M
n N, M xn M
n N, M M (xn). : (xn) M |xn| M n N.
2.1.2 n N . n N.
n N. : n 234 4 n n2 n > 8 xn < xn+1 (xn).
. , , n N n0 N n N, n n0 .
n (xn) (xn), (xn) , , n N . : xn+1 xn, xn+1 > xn,xn u, xn = c
. , (xn) , , , , u, c .
. (1) (1, 23,2 ,2,1,1,1,1, . . . ) ,
.(2) (n2 14n + 8) . , n2 14n + 8 =(n 7)2 41 n N n 7 , , .
n N. n0 N n N, n n0 n0 N, n0 n0 , n N, n n0 .
, n N. n0, n0 N n N, n n0 n N, n n0 .
n0 = max{n0, n0} N.
29
-
n0 n0 , n N, n n0 . , n0 n0 , n N, n n0 . , , n N, n n0 . : , ,
, .
. n2 3n 37 n N, n 157+32
, , n N,n 8. , 2n+1
n+1> 25
13 n N, n > 12 , , n N, n 13.
n2 3n 37 2n+1n+1
> 2513
n N, n max{8, 13} = 13.
, , : , n0 , n0 , , . , , !
. n N .
. (1)n1 > 0 n N, n N., (1)n1 0 n N, n N. n N n N .
n N. n N k N. n N, n k , n N, n > k . k N n N, n > k . , k
N n N, n > k . n N, k N n N, n N, n > k . n N. , , : n N k N
n N,n > k .
n N. , n0 N n N, n n0 . n N, 1 n n0 1, n N. , n N. n0 N n N, n
n0 , , n N, n n0 . : n N n N.
, : n N n N .
.
1. : 1, 4, 9, 16, 25, 36. : 49; 24; ;
30
-
2. (i) (a+b2
+ (1)n1 ab2
).
(ii) m N, (nm[ n
m])+n=1
.: m = 1, 2, 3.
3. (i) , .(ii) .
4. (xn) (yn) (xn + yn).(i) , , .(ii) , .
5. ((1)n1n),( (1)n1
n
),(
18nn2+n+1
),(13n
n!
),(n30
2n
),(2[n
2]),(n 3[n
3])
; ; ;
6. , , , , .
7. (xn) . x xn = x n N.
8. . a, b, p, q, p, q 0. (xn) :
x1 = a, x2 = b xn+2 = pxn+1 + qxn (n N).
n- xn . 1: p 6= 0, q = 0. xn = bpn2 n N, n 2. 2: p = 0, q 6= 0.
xn = aq
n12 n N
xn = bqn22 n N.
3: p 6= 0, q 6= 0. x2 = px+ q.(i) = p2 +4q > 0, , 1 = p+
2, 2 = p
2. ,
+ = a, 1 + 2 = b. xn = 1n1 + 2n1 (n N).(ii) = p2 + 4q = 0, , =
p
2. , = a, + = b.
xn = n1 + (n 1)n1 (n N).(iii) = p2 + 4q < 0 ( q < 0), , 1
=
p+i
2, 2 = pi
2. =
q > 0,
(p2
)2+(
2
)2= 1 , ,
[0, 2) cos = p2, sin =
2
. 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))+n1
sin((n 1)) (n N). n- x1 = x2 = 1 xn+2 = 3xn , xn+2 = xn+1 + xn
,
31
-
xn+2 = 2xn+1xn , xn+2 = xn+1xn . Fibonnaci 1, 1, 2, 3, 5, 8,
13.
9. - - (xn) xn+1 = x1 + + xn (n N); xn+3 = xn+2xnxn+1 (n N).
2.2 .. (xn) x (xn) x x (xn) > 0 n0 N
|xn x| <
n N, n n0 . : (xn) x > 0 |xn x| < . (xn) x
xn x limxn = x limn+
xn = x.
(xn) , (xn) .
: (xn) x n- xn x . : (xn) x n- xn x n .
. xn x, > 0 n0 N - - n N,n n0 () |xn x| < , , |xn x| <
() n N, n n0. :
(n N, n n0) |xn x| <
|xn x| < (n N, n n0).
. (1) ( 1n) 0. 1
n 0.
, > 0. n0 N | 1n 0| < n N, n n0 . n N, | 1n 0| <
1n< n > 1
.
:| 1n 0| < 1
n< n > 1
.
1.1 n0 N, n0 > 1 . , n >1
n0 n N, n n0 . n0 N n N, n n0 n > 1
, , | 1
n 0| < . :
(n N, n n0) n > 1 1n< | 1
n 0| < .
- - n0 N.
32
-
2.1. a 0, n0 = [a] + 1 n N, n > a. a < 0, n0 = 1 n N, n
> a.
. .
: n N, n > 3 1, n N, n > 83 3 = [8
3]+1
n N, n > 2 3 = 2 + 1 = [2] + 1. .
n0 = [1] + 1
n N, n > 1. , n0 N, | 1n 0| < n N,
n n0 .(2) (c) c.
c c.
> 0. n0 N |c c| < n N,n n0 . |c c| < 0 < . , 0 <
n N. n0 = 1 N |c c| < n N, n n0 .(3) ((1)n1) , . - - ((1)n1) x.
> 0 n0 N |(1)n1 x| < n N, n n0 . , n0 , n N, n n0 . n N,n n0
| 1 x| < n N, n n0 |1 x| < . | 1 x| < x (1 ,1 + ) |1 x|
< x (1, 1+). , > 0, x (1 ,1+ ) (1 , 1+ ). , , 0 < 1, ,
.
. (xn) + (xn) + + (xn) M > 0 n0 N
xn > M
n N, n n0 . : (xn) + M > 0 xn > M . (xn) +
xn + limxn = + limn+
xn = + .
, (xn) (xn) (xn) M > 0 n0 N
xn < M
n N, n n0 . : (xn) M > 0 xn < M . (xn)
xn lim xn = limn+
xn = .
33
-
: (xn) + n- xn . : (xn) + n- xn + n . .
6, 7 8 .
. (1) (n) +. n +. M > 0. n0 N n > M n N,n n0 . 1.1, n0 N,
n0 > M . , n > M n0 n N, n n0 . - -
n0 = [M ] + 1 N
, n0 , n > M n N, n n0 .(2) (
n) +.
n +.
M > 0. n0 N n > M n N, n n0.
M > 0, n > M n > M2. :
n > M n > M2 .
n0 N n0 > M2 , , n N, n n0 n > M2, ,
n > M . :
(n N, n n0) n > M2 n > M.
(3) , n < M n < M ,
: n n .
(4) ((1)n1n) (1,2, 3,4, 5,6, . . . ) . - - +. M > 0 n0 N
(1)n1n > M n N, n n0 . n0 , n N, n n0 . n N, n n0 n > M n N,
n n0 n > M . , n > M ,, , ., .
, , .
. (1) ( 1n).
1
na 0 (a > 0).
> 0. n0 N | 1na 0| < n N, n n0 . n N, | 1
na 0| < 1
na< na > 1
n > (1
)1a .
n0 N n0 > (1)1a , , n N, n n0 n > (1)
1a
, , | 1na
0| < .(2) (n) (
n).
na + (a > 0).
34
-
M > 0. n N, na > M n > M 1a . n0 N n0 > M
1a , n N, n n0 n > M
1a , , na > M .
(3) loga n + (a > 1).
M > 0. n N, loga n > M n > aM . n0 N n0 > aM , n N,
n n0 n > aM , , loga n > M .(4) . (a, a2 , a3 , a4 , . . . ),
(an). - : a. a = 1, (1) 1. , a = 0, (0) 0. a 1, (an) a 1, a2 1, a3
1, a4 1, . . . . (an) . , (an) x. > 0 n0 N |an x| < 1 n N, n
n0 . x > an 1 0 n N, n n0 x < an + 1 0 n N,n n0 . x > 0 x
< 0, . , (an) +. M > 0 n0 N an > M n N, n n0 . n N, n n0 .
, (an) . 0 < |a| < 1, (an) 0. > 0. , |an 0| < |a|n <
n > log|a| . n0 N n0 > log|a| , , n N, n n0 n > log|a| , ,
|an 0| < . a > 1, +. M > 0. , an > M n > logaM . n0
N n0 > logaM , , n N, n n0 n > logaM , , an > M .
an
+, a > 1 1, a = 1 0, 1 < a < 1 , a 1
2.4 7 2.5 .
. - . - 2.3(3) 2.23 - , - .
. , lim, limn+ - , . , . limn+ xn , , R . , , limn+ xn = x ( ),
limn+ xn x, limn+ xn = x R .
.
1. , , : limn+ n2n,
limn+ (1)n8n
32n, limn+ log3 n, limn+
(1)n22n3n
.
35
-
2. x limn+ (x+1)2n
(2x+1)n;
3. , : 1n+8
0, 3n+12n+5
32, 1
n+5 0,
n2 7n +, 2n + 2n +, 3+log2 n1+3 log2 n
13.
4. (i) 3+(1)n
2n 0, 3+(1)
n
2n> 0 n N
(3+(1)n2n
) .(ii) (3(1)
n1)n2
+ ((3(1)n1)n
2
) .
5. (i) xn x. > 0 n0() n0 N |xnx| < n N, n n0 . , 0 <
< , n0() n0().(ii) xn +. M > 0 n0(M) n0 N xn > M n N, n n0
. , M > M > 0, n0(M ) n0(M).
6. :(i) (xn) x 0 > 0 |xn x| 0 n N.(ii) (xn) + M0 > 0 xn M0
n N.(iii) (xn) M0 > 0 xn M0 n N.
7. .(i) 0 > 0, xn x , 0 < 0 |xnx| < .(ii) M0 > 0, xn
+ M M0 xn > M .(iii) xn + M - - xn > M .
8. x (xn).(i) , > 0 |xn x| < , |xn x| . . : xn = (1)n1 , x
= 0, = 1. : xn x > 0 |xn x| .(ii) , M > 0 xn > M , M xn M
. . : xn = 1, M = 1. : xn + M > 0 xn M .
9. (xn) x : n0 N > 0 |xn x| < n N, n n0 . ; xn x.
10. (xn) . , xn x, (xn) x .: A (xn) d > 0 A. n0 N |xn x| <
d2 n N, n n0 . n N, n n0 |xnxn0 | |xnx|+|xn0x| < d2+
d2= d.
xn = xn0 n N, n n0 .
36
-
11. (xn) xn Z n N. xn x, (xn) x Z.: 10: - Z 1.
2.3 . , -
.
. > 0. (x , x+ ) - x
Nx() = (x , x+ ).
, x Nx . , Nx x . . . > 0 Nx() x . . . Nx x . . . > 0,
Nx() x . . . .
. > 0. (1,+] - + [,1
)
- .
N+() =(1,+
], N() =
[,1
).
, N . M = 1 , =
1M
(M,+](M > 0) (1
,+] ( > 0) [,M) (M > 0)
[,1) ( > 0).
, x R , Nx() . 1. x Nx(). x Nx().
|xn x| <
xn x , ,
x < xn < x+
, ,xn (x , x+ )
, ,xn Nx().
, xn > M, xn < M
xn , ,
xn (M,+], xn [,M)
37
-
, ,xn N+(), xn N(),
= 1M
., , , .
. x R . xn x > 0 n0 N xn Nx() n N, n n0 , , > 0 xn Nx(). :
xn x Nx x xn Nx .
. , , , . , .
.
1. x R .(i) , 0 < 1 2 , Nx(1) Nx(2).(ii) > 0 n N Nx( 1n)
Nx().
2. x R .(i)
>0Nx() = {x} , ,
x x.(ii)
+n=1Nx(
1n) = {x}.
3. x, y R , x 6= y. > 0 Nx() Ny() = .
2.4 . 2.1. (xn), (yn) . , .
. (xn), (yn) k0,m0 N
xk0 = ym0 , xk0+1 = ym0+1 , xk0+2 = ym0+2 , . . . .
xn a R yn a. > 0. n0 N xn Na() n N, n n0 .
n0 = max{n0, k0} N,
n0 n0 n0 k0 . n0 n0 , xn Na() n N, n n0 .
n0 = n0
+m0 k0 .
n0 k0 , n0 N. n N, n n0 n m0,
yn = xnm0+k0 , nm0 + k0 n0 .
n N, n n0 yn Na(). yn a.
38
-
. (1) (1, 12, 13, 14, 15, 16, . . . ), (2, 5, 1
4, 15, 16, . . . ). -
0. , , , , 0.(2) (xn) . (xn) (x1, x2, x3, . . . ). (x2, x3, x4,
. . . ), (xn+1), (xn). (x3, x4, x5, . . . ), (xn+2). , m N,
xn x R xn+m x R .
: 1n+3
0 log2(n+ 8) +.
. , 2.2 2.3 .
2.2 () , .
2.2. xn x R .(1) x > u, xn > u.(2) x < l, xn < l.(3)
u < x < l, u < xn < l.
. (1) xn x x > u. x u > 0,
|xn x| < x u
, , xn > x (x u) = u.
xn +. M > 0 M u. xn > M , , xn > u.(2) .(3) xn > u
xn < l. xn > u xn < l.
2.3 - (), - - - ., 2.3 - : , . , . 14 2.7 12 2.9.
2.3. (1) xn l n N xn x R, x l.(2) xn u n N xn x R, x u.(3) u
< l xn u n N xn l n N, (xn) .
. (1) x < l, xn < l, xn l n N.(2) .(3) (xn) , u l, l
u.
39
-
. (1) xn x R xn [l, u] n N, x [l, u].(2) a 1, (an) , an 1 n N an
1 n N.(3) ((1)n1n) , (1)n1n 1 n N (1)n1n 1 n N.(4)
(n 3[n
3]) , n 3[n
3] = 0 n = 3k (k N)
n 3[n3] = 1 n = 3k + 1 (k N).
2.4 .
2.4. .
. (xn) . a . 2.2, xn > a , , xn < a. xn > a xn < a.
.
2.5 , , 2.3. - , - - - .
2.5. xn yn n N xn x R, yn y R, x y.
. x > y. a x > a > y. xn > a a > yn . , xn > a
a > yn . xn > yn , , xn yn n N. .
. 1n< 1
n n N 1
n 0, 1
n 0.
, xn < yn n N xn x R ,yn y R , x < y. xn < yn n N xn yn
n N, x y.
2.6 2.7 : .
2.6. xn yn .(1) xn +, yn +.(2) yn , xn .
. (1) M > 0. xn > M , yn xn , xn > M yn xn . yn > M
, ,yn +.(2) .
. (1) n+(1)n1 n1 n N. n1 +, n+ (1)n1 +.(2) n2+2n+1
n+2 n n N n +, n2+2n+1
n+2 +.
(3) [n] >
n 1 n N
n 1 +. [
n] +.
2.7 .
2.7. xn yn zn . xn a zn a, yn a.
40
-
. > 0. |xn a| < , , |zn a| < .
|xn a| < , |zn a| < , xn yn zn .
xn > a , zn < a+ , xn yn zn .
a < yn < a+ , , |yn a| < . yn a.
. (1) 1n (1)
n1
n 1
n n N. 1
n 0 1
n 0,
(1)n1
n 0.
(2) , 1n sinn
n 1
n n N, sinn
n 0.
2.8. , .
. xn x. n0 N |xn x| < 1 n N, n n0 . |xn x| < 1
|xn| = |(xn x) + x| |xn x|+ |x| < 1 + |x|.
|xn| < 1 + |x| n N, n n0 .
M = max{|x1|, . . . , |xn01|, 1 + |x|}
|xn| M n N, 1 n n0 1 |xn| < 1 + |x| M n N, n n0 . |xn| M n N,
(xn) .
. 2.8. H ((1)n1) - .
2.9. (1) +, .(2) , .
. (1) xn +. n0 N xn > 1 n N,n n0 .
l = min{x1, . . . , xn01, 1},
xn l n N, 1 n n0 1 xn > 1 l n N, n n0 . xn l n N, (xn) . , M
> 0 xn > M . M > 0 (xn), (xn) .(2) .
. (1), (2) 2.9 . -
((1+(1)n1)n
2
), (1, 0, 3, 0, 5, 0, 7, . . . ), .
, + (1+(1)n1)n
2 0 n N.
, (1, 0,3, 0,5, 0,7, . . . ) , .
(xn) (xn). 2.10 .
2.10. xn x R , xn x.
41
-
. xn x. > 0. |xn x| < .
|(xn) (x)| = |xn x| < .
xn x. xn +. M > 0. xn > M , xn < M ,, xn = (+)., , xn ,
xn + = ().
(xn) (yn) (xn+ yn). 2.11 .
2.11. xn x R yn y R x+ y , xn + yn x+ y.
. xn x, yn y. > 0. |xn x| < 2 |yn y| < 2 . |xn x| 0. M
> 0 M M l. xn > M , xn > M l, xn+yn > (M l)+ l =M .
xn + yn + ={
(+) + y(+) + (+)
}.
.
. (1) 1n 0 (1)
n
n 0, 1
n+ (1)
n
n 0 + 0 = 0.
(2) n 1n 0, n2+1
n= n+ 1
n () + 0 = .
(3) n + n +, n+
n (+) + (+) = +.
(+)+ (), + R :
. (1) n+ c +, n (n+ c) + (n) = c c.(2) 2n +, n 2n+ (n) = n +.(3)
n +, 2n n+ (2n) = n .(4) n+ (1)n1 +, n (n+ (1)n1) + (n) = (1)n1
.
(xn) (yn) (xn yn). (xn yn) , xnyn = xn+(yn). 2.12 .
2.12. xn x R yn y R x y , xn yn x y.
(xn) (yn) (xnyn). 2.13 .
42
-
2.13. xn x R yn y R xy , xnyn xy.
. xn x, yn y. > 0. |xn x| < 3|y|+1 , , |yn y| < min{
3|x|+1 ,
13}. |xn x| < 3|y|+1 |yn y| 0. xn > Ml , , yn > l. xn
>
Ml
yn > l.
xnyn > Ml l =M , , xnyn + ={
(+)y(+)(+)
}.
.
. (1) 1n 0 (1)
n
n 0, (1)
n
n2= 1
n(1)nn
0 0 = 0.(2) n1
n 1 1
n 0, n1
n2= n1
n1n 1 0 = 0.
(3) n + 1n , nn2 = n(1n) (+)() = . (nn2) .(4) c xn x R cx , , ,
c = 0 x = . , c c,
cxn cx.
, c = 0, , x R , cxn = 0xn = 0 0.(5) a > 0. c > 0, cna
c(+) = +. c < 0, cna c(+) = .(6) a > 0, cna c0 = 0.(7) n.
a0+a1x+ +akxk , ak 6= 0, k 1.
a0 + a1n+ + aknk = aknk(a0ak
1nk
+ a1ak
1nk1
+ + ak1ak
1n+ 1
).
1, 0. ,akn
k ak(+).
a0 + a1n+ + aknk ak(+)1 =
{+, ak > 0, ak < 0
., limn+(a0 + a1n+ + aknk) = limn+ aknk . : 3n2 5n+ 2 + 1
2n5 + 4n4 n3 .
(8) . a
43
-
(1 + a+ a2 + + an1 + an), (1 + a, 1+ a+ a2 , 1+ a+ a2 + a3 , . .
. ). :
1 + a+ a2 + + an
+, a 1 1
1a , 1 < a < 1 , a 1
. a > 1, 1 + a+ a2 + + an = an+11
a1 (+)1a1 = +.
a = 1, 1 + a+ a2 + + an = n+ 1 +. 1 < a < 1, 1 + a+ a2 + +
an = an+11
a1 01a1 =
11a .
a 1. : an+1 = 1 + (a 1)(1 + a + a2 + + an) n N. 1 + a +a2 + + an
x R, an+1 1 + (a 1)x. , (an+1) . : 1 + a + a2 + + an = an+11
a1 2
1a n N 1 + a + a2 + + an = an+11
a1 0 n N. 0 0. l 0 < l < x. xn > l. > 0. |xn x| <
lx. xn > l |xn x| < lx. , 1xn 1x = |xnx|xnx < lxlx = ,
1
xn 1
x.
xn +. > 0. xn > 1 , 0 1, 1loga n 1
+ = 0.
(2) xn 0. , (1)n1
n 0, n
(1)n1 = (1)n1n
. 10 .
. , ; 2.16.
2.16. xn 6= 0 n N xn 0.(1) xn > 0, 1xn +.(2) xn < 0, 1xn
.
. (1) M > 0. |xn 0| < 1M . xn > 0, |xn 0| < 1M xn
> 0. 0 < xn M , 1
xn +.
(2) .
(xn) (yn) (xnyn
). (xn
yn)
, xn
yn= xn
1yn
. 2.17 .
2.17. yn 6= 0 n N. xn x R yn y R xy , xn
yn x
y.
. (1) n. a0+a1x++akxkb0+b1x++bmxm ,
ak 6= 0, bm 6= 0.
a0+a1n++aknkb0+b1n++bmnm =
aknk
bmnm
a0ak
1
nk+
a1ak
1
nk1++
ak1ak
1n+1
b0bm
1nm
+b1bm
1nm1
++ bm1bm
1n+1
, , 1. akn
k
bmnm= ak
bmnkm ,
a0 + a1n+ + aknk
b0 + b1n+ + bmnm
akbm(+), k > m
akbm, k = m
0, k < m
. : n32n2+n+1
2n23n1 +,n2+nn+2
, n4n3n4+1
1 n2+n+4n3+n2+5n+6
0.(2) 2n3+n2+n+1
2n+3 ,
(2n3+n2+n+12n+3
)7 ()7 = .(3) n3+n+73n3+n2+1
13,
(n3+n+73n3+n2+1
)3 (13)3 = 1
27.
, 00 +
+ , 0 0 + + R, , [0,+], , :
45
-
. (1) cn 0, 1
n 0 ( c
n)/( 1
n) = c c.
(2) 1n 0, 1
n2 0 ( 1
n)/( 1
n2) = n +.
(3) 1n2
0, 1n 0 ( 1
n2)/( 1
n) = 1
n 0.
(4) 1n 0, 1
n2 0 ( 1
n)/( 1
n2) = n .
(5) 1n
2+(1)n1
n 3
n n N, 2+(1)
n1
n 0. , 1
n 0
(2+(1)n1
n)/( 1
n) = 2 + (1)n1 , 2 + (1)n1 1 n N
2 + (1)n1 3 n N.(6) (1) c > 0, (2), (3) (5) (xn) (yn) + [0,+]
.
(xn) (|xn|). 2.18 .
2.18. xn x R , |xn| |x|.
. xn x. > 0. |xn x| < . |xn| |x| |xn x| < , , |xn| |x|.
xn + xn . M > 0. xn > M xn < M ,. , |xn| > M , |xn| + =
| |.
. 2.18 . ,(1)n1 = 1 1
(1)n1 .
- - .
. (1) a > 0.
na 1 (a > 0).
a = 1 : n1 = 1 1.
a > 1. : > 0. | n
a1| < ( n
a > 1) n
a1 <
na < 1+ 1
n< loga(1+) n >
1loga(1+)
. n0 N n0 > 1loga(1+) . n N, n n0 n >
1loga(1+)
,, | n
a 1| < . n
a 1.
: Bernoulli
(1 + a1n)n 1 + na1
n= a
, ,1 n
a 1 + a1
n
n N. na 1.
0 < a < 1, 1a> 1, n
a = 1
n
1/a 1
1= 1.
(2) nn 1.
46
-
Bernoulli,
(1 +n1n
)n 1 + nn1n
=n ,
1 n
n (1 +
n1n
)2 < (1 + 1n)2
n N. , , nn 1.
(3) a > 1.
an
n + (a > 1).
: Bernoulli,
(a)n = (1 +
a 1)n 1 + n(
a 1) > n(
a 1)
, ,an
n> n(
a 1)2
n N. ann +.
: b 1 < b < a. Bernoulli
bn = (1 + b 1)n 1 + n(b 1) > n(b 1)
n N. an
n= b
n
n(ab)n > (b 1)(a
b)n
n N , ab> 1, an
n +.
(4) (an) - a > 1, |a| < 1. a > 1. : b = a 1 > 0.
Bernoulli,
an = (1 + b)n 1 + nb > nb
n N. nb +, an +. : an > nb. M > 0. , an > M nb > M n
> M
b. n0 N
n0 > Mb . n N, n n0 n >Mb
, , an > M . an +. : |a| < 1, : 0 < 1
a< 1, 1
an= ( 1
a)n 0 , , an +.
|a| < 1. a = 0 , 0 < |a| < 1. : 1|a| > 1, , ,
1|a|n = (
1|a|)
n +. |a|n 0. ,
|a|n an |a|n
n N, an 0. : b = 1|a| 1 > 0. Bernoulli,
|a|n = 1(1+b)n
11+nb
< 1nb
47
-
n N. 1nb< an < 1
nb
n N, an 0. : > 0. |a|n < 1
nb |an 0| <
1nb< n > 1
b. n0 N n0 > 1b .
n N, n n0 n > 1b , , |an 0| < . an 0.
., 2.19, 4.3.
, - - . 1.3 ab . 00 , 1+ , 1 ,(+)0 , 0 .
(xn) (yn) (xn
yn).
2.19. xn > 0 n N. xn x R yn y R xy , xnyn xy . , xn 0 yn ,
xn
yn +.
. (1) 2.19, na 1 a > 0.
, (a) ( 1n). a a 1
n 0,
na = a
1n a0 = 1.
(2) (an) a > 1, 0 < a < 1 , , 2.19. (a) (n). a a n +,
an a+ , +, a > 1, 0, 0 < a < 1.(3) n
n 1 2.19. (n)
( 1n), n + 1
n 0, (+)0 .
(+)0 00 , - + 0 0 0 [0,+] .
. (1) n +, 1n 0 n 1n = n
n 1.
(2) a > 1, an +, 1n 0 (an) 1n = a a.
(3) a > 1, an +, 1n 0 (an) 1n = 1
a 1
a.
(4) nn +, 1n 0 (nn) 1n = n +.
(5) nn +, 1n 0 (nn) 1n = 1
n 0.
(6) nn +, (1)n1
n 0 (nn)
(1)n1n = n(1)
n1 . , n(1)
n1= n 1 n N n(1)n1 = 1
n 1
2 n N.
(7) , (xn) . xn 0, yn 0 (xnyn) [0,+] .
2.19 0 = +, , , . 0 .
48
-
. (1) 1n 0, n ( 1
n)n = nn +.
nn n n N.(2) 1
n 0, 2n 1 ( 1
n)2n1 = n2n+1 .
(3) (1)n
n 0, n
( (1)nn
)n= (1)n2nn , (1)n2nn =
nn 4 n N (1)n2nn = nn 1 n N.
, 1+ 1 , - 2.5.
.
1. 2.2, 2.3, 2.6, 2.9, 2.10, 2.11, 2.12, 2.13, 2.15,2.16
2.17.
2. ((n+1)27(n+3)79
(2n+1)106
),(n2+(1)nn+ 1
n
3n+2(1)n1n
),(n(n+1)n+4
4n34n2+1
), ((1
n)5 + n4),(( n
3+n+13n2+3n+1
)9),( 3n+(2)n3n+1+2n+1
), (n2 + n+ 1
n2 + 1).
3. - - (1 2+ 22 + + (1)n2n), (1+ 2+ 22 + + 2n),
(1 + 1
2+ + 1
2n
),(27
37+ 2
8
38+ + 2n+6
3n+6
),(2n
3n+ 2
n+1
3n+1+ + 22n
32n
).
4. xn 6= 1 n N, x 6= 1. xn x xn1+xn x
1+x.
5. (xn) - : xn+1 = xn+2, xn+3 = xn3, xn+1 = xn23, xn+2 =
xn2+3,xn+1 = xn
2 + 3, xn+2 = xn+1 + xn3 ;
6. (xn)(i) 1 < xn n
2+3nn2+1
.(ii) log10 n2
2 log10 n+4< xn 00, x = 0, x < 0
[nx] [ny]
+, x > y0, x = y, x < y
nx [ny]
+, x > y 0, x = y Z , x < y , x = y R \ Z
10. : 22n+(1)n1n 0,(12+ (1)
n1
4
)n 0.49
-
11. n
k=1n
n2+k 1
nk=1
1n2+k
1.: n n
n2+n
nk=1
nn2+k
n nn2+1
n N.
12. n2x2n 2n(n 1)xn + n2 2n 3 0, xn 1.
13. 0 < a xnn b, xn 1.
14. 0 a b c. nan + bn b, n
an + bn + cn c.
: bn an + bn 2bn .
15. : nn3 1, n
n4 + 3n2 + n+ 1 1.
16. 1nn
nk=1 k
k 1.
17. , :(i) n5 + 4n3 < 100;(ii) n7 35n6 + n3 47n < 84 n
N;(iii) 3
2< 7n
3n+54n3+n2+35
< 2;(iv) 2n4n3+7n3+n2+3 78;(v) 2n3n2+7n+1
n3+n2+3 1 n N;
18. 2.3, (2(1)
n1),((1 + (1)
n1
2
)n), ((1)n1 + 10n3
),((1)n1 n
n+1
).
19. |xn| 0, xn 0 : .
20. xn x yn y, x, y R, max{xn, yn} max{x, y} min{xn, yn} min{x,
y}.
21. (i) : n 1n= 1
n+ + 1
n 0 + + 0 = 0.
(ii) : (1 + 1n)n = (1 + 1
n) (1 + 1
n) 1 1 = 1.
2.11, 2.13, 2.19;
22. xn x R yn y R . x < y, xn < yn .
23. , xn [l, u] n N xn x, x [l, u]. x (xn), xn (l, u) n N; ;
24. (i) xn + (yn) , xn + yn +.(ii) xn (yn) , xn + yn .(iii) xn 0
(yn) , xnyn 0.(iv) xn + yn > l > 0, xnyn + , .(v) xn + yn
< u < 0, xnyn +, .
50
-
25. (i) a < 1 |xn+1| a|xn|, xn 0.: |xn+1| a|xn| n n0 ,
|xn|
|xn0 |an0
an n n0 .(ii) a > 1 xn+1 axn > 0, xn +.(iii)
xn+1xn
a < 1, xn 0.(iv) xn+1
xn a > 1, xn + xn .
(v) , a > 1, ann +. , (n!)
2
(2n)! 0 2nn!
nn 0.
26. (i) (xn), (yn) (xn + yn) .(ii) (xn), (yn) (xnyn) .
27. (i) (xn + yn) (xn), (yn) , , , .(ii) (xnyn) (xn), (yn) , , ,
.
28. (xn), (yn) xn, yn > 0 n N, xn 0, yn + (xnyn) .
29. x1 > 0 xn+1 x1 + + xn n N. 0 < a < 2, xnan
+. (2n) a = 2.
30. (i) x (rn) rn Q n N rn x.: , n N rn Q x 1n < rn 0. M {xn
|n N}, n0 N xn0 > M . (xn) ,
xn xn0 > M
n N, n n0 . xn +. (xn) . x = sup{xn |n N} xn x. > 0. x <
x, x {xn |n N}. n0 N x < xn0 . (xn) ,
x < xn0 xn
n N, n n0 . x {xn |n N},
xn x < x+
n N. x < xn < x+ n N, n n0 , , xn x.(2) .
2.1. (xn) , , 2.1, (xn) , x, . , xn x n N. , , (xn) , xn <
xn+1 x , ,xn < x n N. .: (xn) xn x, xn x n N. , , (xn) , xn <
x n N. (xn) xn x, xn x n N. , , (xn) , xn > x n N.
2.1 . - , . : , , , , n- . 2.1 (, ), .
52
-
2.1 . , - , - - . , , 5 2.4.
2.1 Supremum R ,, . 2.1 , , Supremum. , Supremum . 12.
. (xn) x1 = 1 xn+1 =2xn n N.
(xn) 1,2 ,
22 , . . . .
. , x1 x2 xn xn+1 n N. 0 1 . : xn xn+1 2xn 2xn+1
2xn
2xn+1
xn+1 xn+2 . xn xn+1 n N, (xn) , , . xn xn+1 xn
2xn , xn 2
n N. (xn) , , . xn x. xn+12 = 2xn n N, x2 = 2x, x = 0 x = 2.
(xn) x1 = 1, xn 1 n N. x 1 ,, x = 2. (xn) . xn xn+1 xn
2xn ( xn 0) xn 2. ,
xn 2 n N, (xn) 2. . x1 2 . xn 2 n N. xn+1 =
2xn
2 2 = 2,
xn 2 n N.
2.20. ((1 + 1n)n) .
. (1+ 1n)n < (1+ 1
n+1)n+1 (n+1
n)n < (n+2
n+1)n+1
nn+1
(n+1n)n+1 < (n+2
n+1)n+1 n
n+1< ( n
2+2nn2+2n+1
)n+1 nn+1
< (1 1n2+2n+1
)n+1
Bernoulli. ,
(1 1n2+2n+1
)n+1 > 1 n+1n2+2n+1
= 1 1n+1
= nn+1
n N. (1 + 1
n)n < 4 , , 1
2 1 nn+1
n
n
= 1n(n+ 1
n) = 1
n
n+1+n> 1
n
2n= 1
2
n N.
. 2.1 2.20, ((1 + 1n)n) ,
e. ,
e = limn+
(1 +
1
n
)n.
53
-
((1 + 1n)n) , (1 + 1
n)n < e n N.
e . 8.8.
. e y > 0
log y ln y
loge y.
2.21 , , 1.6.
2.21. (1) log(yz) = log y + log z y, z > 0.(2) log y
z= log y log z y, z > 0.
(3) log(yz) = z log y y > 0 z.(4) loga y =
log ylog a y > 0 a > 0, a 6= 1.
(5) log 1 = 0, log e = 1.(6) 0 < y < z, log y < log
z.
1+ 1 , 1 + 1 [0,+] .
. (1) 1 1, n + 1n = 1 1.(2) a > 1 b = log a. 1 + 1
n 1, bn + (1 + 1
n)bn = ((1 + 1
n)n)b eb = a.
(3) nn 1, n + ( n
n)n = n +.
(4) (xn) (yn) xn 1, yn + (xnyn) [0, 1].
(5) 1nn n(1)n1
n nn n N, n
(1)n1n 1. , n + (
n(1)n1
n
)n= n(1)
n1 , n(1)n1 = 1n 1
2 n N
n(1)n1
= n 1 n N.(6) (xn) (yn) xn 1, yn (xn
yn) [0,+] .
.
. (1) (xn), xn = 1 + 11! +12!+ + 1
n!(n N).
(xn) , ,
1 +1
1!+
1
2!+ + 1
n! e.
xn+1 = 1 +11!+ 1
2!+ + 1
n!+ 1
(n+1)!= xn +
1(n+1)!
> xn
n N, (xn) . k! 2k1 k N. k = 1 , k N, k 2 k! = 1 2 3 k 1 2 2 2 =
2k1 . ,
xn = 1 +11!+ 1
2!+ + 1
n! 1 + 1
20+ 1
21+ + 1
2n1= 1 +
1( 12)n
1 12
< 1 + 11 1
2
= 3
54
-
n N. (xn) , , , , ., tn =
(1 + 1
n
)n(n N). Newton (
10 1.3),
tn = 1 +(n1
)1n+(n2
)1n2
+ +(nk
)1nk
+ +(nn
)1nn
= 1 + 11!+ 1
2!(1 1
n) + + 1
k!(1 1
n)(1 2
n) (1 k1
n)
+ + 1n!(1 1
n) (1 n1
n).
> 0 < 1,
tn 1 + 11! + +1n!
= xn
n N. k, n N, 1 k n, () k-,
tn 1 + 11! +12!(1 1
n) + + 1
k!(1 1
n)(1 2
n) (1 k1
n).
n +,
e 1 + 11!+ 1
2!+ + 1
k!= xk
k N , , e xn n N.
tn xn e
n N , tn e, xn e. 8 10.(2) (xn), xn = 1 + 12 +
13+ + 1
n(n N).
1 +1
2+
1
3+ + 1
n +.
xn+1 xn = 1n+1 > 0 n N, (xn) .
x2n xn = 1n+1 + +1
n+n 1
n+n+ + 1
n+n= n
n+n= 1
2
n N. ,
x2 x1 > 12 , x22 x2 >12, x23 x22 > 12 , . . . , x2k1
x2k2 >
12, x2k x2k1 > 12 .
, x2k x1 > k2 ,
x2k >k2+ 1
k N. (xn) . , u xn u n N, x2k u k N. k2 + 1 u ,, k 2u 2 k N. .
(xn) . xn +. 5 2.7 1 2.8.(3) (xn), xn = 1 + 122 +
132
+ + 1n2
(n N). (xn) . xn+1 xn = 1(n+1)2 > 0 n N, (xn) .
xn 1 + 112 +123 + +
1(n1)n = 1 +
11 1
2+ 1
2 1
3+ + 1
n1 1n= 2 1
n< 2
n N. (xn) , , . 2.8.
55
-
, , 0.
. (an) (bn) an bn n N. , [a1, b1], [a2, b2], [an+1, bn+1] [an,
bn] n N. :(i) (an) (bn) .(ii) x an x bn n N.(iii) x (ii) bn an 0. ,
(an), (bn) x .
. (bn) , an bn b1 n N, (an) , , . (an)
an a.
, (an) , a1 an bn n N, (bn) , , . (bn)
bn b.
an bn n N, a b.
an a b bn
n N. x [a, b] an a x b bn n N. , x an x bn n N, a x b, x [a, b].
x an x bn n N [a, b]. , x a = b , ,bn an 0. x x = a = b.
(ii) , , 13.
. (1) , , . 1 , n N, n 2, 2n - 2n . pn qn , , . ,, (pn)+n=2,
(qn)
+n=2 .
p2 = 42, q2 = 8 - -
pn+1 = 2pn(2 +
(4 pn2
4n
) 12) 1
2 , qn+1 = 4qn(2 +
(4 + qn
2
4n
) 12)1
(n N, n 2)
qn = pn(1 pn2
4n+1
) 12 (n N, n 2).
56
-
(pn) (qn) ,
pn+1 = 2pn(2 +
(4 pn2
4n
) 12) 1
2 > 2pn(2 + 4
12
) 12 = pn
qn+1 = 4qn
(2 +
(4 + qn
2
4n
) 12)1
< 4qn(2 + 4
12
)1= qn
n N, n 2. ,
qn = pn(1 pn2
4n+1
) 12 > pn
n N, n 2. , (pn), (qn) pn p qn q. qn = pn
(1 pn2
4n+1
) 12 n N, n 2,
q = p(1 p20
) 12 = p.
qnpn 0, , , x pn x qn n N, n 2. x : p = q = x. , , , 2, : pn 2
qn n N, n 2.
limn+
pn = limn+
qn = 2.
. 20 7.3.(2) p- . x [0, 1) p N, p 2.
xn = [pnx] p[pn1x]
n N. xn Z n N. [pn1x] pn1x < [pn1x] + 1 p[pn1x] pnx <
p[pn1x] + p p[pn1x] [pnx] < p[pn1x] + p, 0 xn < p , xn Z,
0 xn p 1.
, n N xn 0, 1, . . . , p 1.
sn =x1p+ + xn
pn, tn =
x1p+ + xn
pn+ 1
pn(n N).
sn+1 = sn +xn+1pn+1
sn , tn+1 = tn + xn+1pn+1 +1
pn+1 1
pn tn + p1pn+1 +
1pn+1
1pn
= tn
n N. (sn) (tn) . ,
sn tn
n N, - . ,
tn sn = 1pn 0,
57
-
(sn), (tn) . ;
sn =([px]p
[x])+([p2x]p2
[px]p
)+ +
([pn1x]pn1
[pn2x]pn2
)+([pnx]pn
[pn1x]pn1
)= [p
nx]pn
[x] = [pnx]pn
.
, [pnx] pnx < [pnx] + 1
sn x < sn + 1pn = tn .
sn x tn x.
. (sn) p- ( ) x (tn) p- x. (xn) p- x.
p- x [0, 1) : p 1. - - n0 N xn = p 1 n N, n n0 .
tn+1 = tn +xn+1pn+1
+ 1pn+1
1pn
= tn +p1pn+1
+ 1pn+1
1pn
= tn
n N, n n0 . (tn) , tn x, tn = x. sn x < tn n N.
: p = 2 0, 1, p = 3 0, 1, 2 , , p = 10 0, 1, . . . , 9.
p- 8.
.
1. 2.1.
2. (i) : (1 + 1n)n+3 e, (1 + 1
n+2)3n+5 e3 , (1 1
n)n 1
e, (1 + 2
n)n e2 ,
(1 2n)n 1
e2.
(ii) (1 + kn)n ek k Z.
3. ((1 + 1
n)n+1
) , e
(1 + 1n)n+1 > e n N.
4. x1 = 1 xn+1 = xn + 1xn2 n N. (xn) .
5. 7xn+1 = xn3 + 6 n N. , x1 , (xn) .
6. x1 > 0 xn+1 = 6+6xn7+xn n N. , x1 , (xn) .
7. (i) a > 1, (an) , an+1 = aan (n N), an +. 0 < a <
1.(ii) a > 1, (an
n) ,
58
-
an+1n+1
= ann+1
an
n(n N), an
n +.
(iii) a > 1, ( na) ,
(
2na)2
= na (n N), n
a 1.
a = 1, 0 < a < 1;
8. (i) a, x1 > 0 xn+1 = 12(xn +
axn
) n N, (xn)
, , .(ii) xn, yn Z n N, x1 = y1 = 1 xn+1 + yn+1
2 = (xn + yn
2)2
n N, xnyn
2.
9. (xn) 2xn+1 xn+xn+2 n N. yn = xnxn+1 n N. (yn) yn 0.
10. (i) 0 < x1 y1 xn+1 =xnyn yn+1 = xn+yn2 n N,
(xn) , (yn) , xn yn n N (xn),(yn) .(ii) 0 < x1 y1 xn+1 =
2xnynxn+yn yn+1 =
xnyn n N,
(xn) , (yn) , xn yn n N (xn),(yn) .
11. I f : I R x I 0 > 0 f(x) f(x) f(x) x, x (x 0, x+ 0) I , x
< x < x . f I .: a, b I a < b, f(a) > f(b). f(a) >
f(a+b
2), a1 =
a, b1 =a+b2
, f(a+b2) > f(b), a1 = a+b2 , b1 = b. f(a1) > f(b1).
, [a1, b1], [a2, b2], . . . [an+1, bn+1] [an, bn], f(an) >
f(bn) bn an = ba2n n N. an bn n N an , bn . 0 , .
12. . Supremum.: . 1
2n 0. , - A. x1 A
y1 A. [x1, y1] A A. x1+y1
2 A, x2 = x1, y2 = x1+y12 , , x2 =
x1+y12
,y2 = y1 . [x2, y2] A A. , [x1, y1], [x2, y2], . . . [xn+1,
yn+1] [xn, yn] ynxn = y1x12n1 n N [xn, yn] an A un A. u xn u, yn u
, , an u, un u. u A.
13. (an) (bn) an bn n N. {an |n N} {bn |n N} , , x an x bn n
N.
59
-
14. A - R - - a : N A -- . an = a(n), A - A = {an |n N} , , A .
, A - . I ( ) .: I = {an |n N}. [x1, y1] I y1x1 > 0 a1 / [x1,
y1]. [x2, y2] [x1, y1] y2 x2 > 0 a2 / [x2, y2]. , [x1, y1], [x2,
y2], . . . [xn+1, yn+1] [xn, yn] an / [xn, yn] n N. [xn, yn] n N ,
, 6= an n N. .
2.6 Supremum, infimum . A. (xn) A xn A
n N, (xn) A. 2.22 supremum -
infimum - : supA A infA A.
2.22. - A.(1) A supA A supA.(2) A infA A infA.
. (1) A , x = supA . n N x 1
n A, xn A
x 1n< xn x.
(xn) A xn x., A , supA = +. n N A, n N xn A
xn > n.
(xn) A xn +., (xn) A. (xn) , supA, xn supA n N. A supA.(2) .
.
1. 2.22.
60
-
2. [0, 2], [0, 2), {2}, [0, 1] {2} supremum . , ( ) , , . ,, ,
.
3. N, Z, Q, { 1n|n N} ,
supremum infimum .
4. - A u A. u = supA A u. infA l A.
5. - A. supA A A supA. supA / A, A supA. infA.
6. f : [0, 1] R [0, 1] f(0) > 0 f(x) 6= x x [0, 1]. A = {x
[0, 1] | f(x) > x}, supA A f(1) > 1.
2.7 .. (xn). n1 , n2 , n3 , . . . , nk , . . . n, , n1 < n2
< < nk < nk+1 < . (xn). x1 , x2 , . . . , xn , . . .
xn1 , xn2 , . . . , xnk , . . .. : xn1 , xn2 . , (xnk). , (xnk)
(xn). , (xnk) x n : N R (x n)(k) = x(n(k)) = xn(k) = xnk , n : N N
x : N R, , , (nk) (xn).
, n1 < n2 < < nk < nk+1 < , .
. (1) n1 = 1, n2 = 3, n3 = 10, n4 = 11, (xn) x1 , x3 , x10 , x11
.(2) n1 = 2, n2 = 5, n3 = 6, n4 = 9, n5 = 13, (xn) x2 , x5 , x6 ,
x9 , x13 .(3) , n1 = 2, n2 = 5, n3 = 6, n4 = 10, n5 = 8 (xn). x2 ,
x5 , x6 , x10 , x8 (xn) : x10 x8 (xn) - x9 - x10 x8 .
. .(1) nk = 2k (k N), (x2k) (x2, x4, x6, x8, x10, . . . )
61
-
(xn).(2) nk = 2k 1 (k N), (x2k1) (x1, x3, x5, x7, x9, . . . )
(xn).(3) nk = k (k N), (xk) (x1, x2, x3, x4, x5, . . . ), (xn). ,
(xn) (xn).(4) nk = 2k1 (k N), (x2k1) (x1, x2, x4, x8, x16, . . .
).(5) nk = k2 (k N), (xk2) (x1, x4, x9, x16, x25, . . . ).
(xnk) k. k 1, 2, 3, . . . , nk (xn).
2.2. nk N nk < nk+1 k N. nk k k N.
. n1 1 n1 N. nk k k N. nk+1 > nk nk, nk+1 N, nk+1 nk + 1,
nk+1 k + 1. nk k k N.
2.23. , .
. xn x R (xnk) (xn). xnk x. > 0. n0 N xn Nx() n N, n n0 . , k
N, k n0 nk n0 , , 2.2, nk k. k N, k n0 xnk Nx(). xnk x.
2.23 , , : , . - 14 12 2.9.
. ((1)n1) . , (1)(2k1)1 = 1 1 (1)(2k)1 = 1 1.
2.24 .
2.24. x R x2k x, x2k1 x. xn x.
. x R x2k x, x2k1 x. > 0. k0 N x2k Nx() k N, k k0 . , k0 N
x2k1 Nx() k N, k k0 .
n0 = max{2k0, 2k0 1}.
n0 N xn Nx() n N, n n0 .
.
. (xn) xn = 1 12 +13 1
4+ + (1)n1 1
n(n N).
(xn) .
x2k+2 x2k = 12k+1 1
2k+2> 0
62
-
k N. ,
x2k = 1 (12 13) (1
4 1
5) ( 1
2k2 1
2k1)12k< 1
k N, . (x2k) , , .
x2k+1 x2k1 = 12k +1
2k+1< 0
k N. ,
x2k1 = (1 12) + (13 1
4) + + ( 1
2k3 1
2k2) +1
2k1 > 0
k N . (x2k1) , , .,
x2k x2k1 = 12k 0,
(x2k), (x2k1) . (xn) . 2 2.8.
. . , ((1)n1) . , ((1)n1), , : 1. ((1)n1) . , .
Bolzano - Weierstrass. .
. (xn) l, u l xn u n N. (xn) , : , , . 1. [l, u] [l, l+u
2], [ l+u
2, u]. ()
(xn) [l, u], (xn). [l1, u1]. -, [l, l+u
2] (xn) [ l+u2 , u]
, [l1, u1] [l, l+u2 ]. [l+u2, u] [l, l+u
2]
, [l1, u1] [ l+u2 , u]. , , , [l1, u1] . [l1, u1] [l, u], u1 l1
= ul2 [l1, u1] (xn). (xn) ( ) [l1, u1]: xn1 [l1, u1]. 2. [l1, u1]
[l1, l1+u12 ], [
l1+u12, u1]. [l1, u1]
(xn), (xn). [l2, u2] - . [l2, u2] [l1, u1], u2 l2 = u1l12 [l2,
u2] (xn). (xn) ( ) [l2, u2]: xn2 [l2, u2]. , , n2 > n1. ,
63
-
(xn) [l2, u2]. 3. [l2, u2] [l2, l2+u22 ], [
l2+u22, u2].
[l2, u2] (xn), (xn). [l3, u3]. [l3, u3] [l2, u2], u3 l3 = u2l22
[l3, u3] (xn). (xn) ( ) [l3, u3]: xn3 [l3, u3]., , n3 > n2 . .
[lk, uk](k N)
[lk+1, uk+1] [lk, uk], uk+1 lk+1 = uklk2
k N. , xnk (k N) (xn) nk+1 > nk
xnk [lk, uk]
k N. uk+1 lk+1 = uklk2 k N
uk lk = ul2k
k N, uk lk 0.
, (lk), (uk) . lk x uk x. , nk+1 > nk k N, (xnk) (xn) ,
lk xnk uk
k N, xnk x.
8 Bolzano - Weierstrass. + .
. , (1, 0, 3, 0, 5, 0, 7, . . . ) +. , +, +: , (1, 3, 5, 7, . .
. ). .
2.25. (1) +.(2) .
. (1) (xn) . +. . 1. (xn) , > 1. : xn1 > 1. 2. (xn) , >
2. > 2; - -
64
-
(xn) 2. n0 N xn 2 n N, n n0 .
u = max{x1, . . . , xn01, 2}
xn u n N, 1 n n0 1 xn 2 u n N, n n0 . xn u n N, u (xn) . , ,
(xn) 2: xn2 > 2. ,, n2 > n1. > 2. 3. (xn) , - - > 3.
(xn) 3: xn3 > 3. , , n3 > n2. . xnk (k N) (xn) nk+1 >
nk
xnk > k
k N. (xnk) (xn) xnk +.(2) .
2.26 2.25 Bolzano - Weierstrass.
2.26. .
. (xn) , . (xn) , +. , (xn) , .
.
1. 2.25.
2. a < b < c < d. a, b, c d.
3. (xn) : () , (-) , , , . (xnk) (xn) .
4. x R x3k x, x3k1 x, x3k2 x. 2.24, xn x.
5. (i) (xn) (xnk) xnk x R. xn x.(ii) (xn) (xnk) xnk x R. xn
x.(iii) xn = 1 + 12 +
13+ + 1
n(n N). 2.5
x2k k2 + 1 k N. xn +.(iv) (1+ a1)(1+ a2) (1+ an) 1+ a1+a2 + + an
n N a1, . . . , an 0.(v) b . a = b, a > b, a < b, limn+
(a+1)(a+2)(a+n)(b+1)(b+2)(b+n) .
65
-
6. x1 > 0 xn+1 = 1 + 21+xn n N.(i) (x2k), (x2k1) .(ii) (xn)
.
7. a, b, x R , a 6= b. x2k a, x2k1 b (xnk) xnk x.(i) (xnk) (x2k)
(x2k1).(ii) x = a x = b.
8. (i) (xn). xn - m N, m > n xm > xn . , (xn) -, (xn) ,
(xn) -, .(ii) (i) Bolzano - Weierstrass , , 2.26.
9. (xn) n N (xnk) k N.
10. , .
11. (xn) x R (xn) x.
12. xn < x n N. sup{xn |n N} = x (xn) x.
13. (xn) (xnk). (xnk) (xn).
14. (i) (xn) . 2.23.(ii) (xn) l, u, u < l xn u n N xn l n N.
2.3(3).
15. (i) xn x xn 6= x n N. (xn) .(ii) (rn) , rn =
qnpn
, qn Z, pn N n N. pn +.(iii) x (rn) rn x rn = qnpn , qn Z, pn N
n N. qn + pn +, x > 0, pn , x < 0.
16. (i) 2 2.5, (1 +12n)n e 12 , (1 + 2
3n)n e 23 , , (1 + r
n)n er r Q.
x.(ii) > 0. ex s, t Q, s < x < t
66
-
ex < es < et < ex + .(iii) (i), ex < (1+ s
n)n < (1+ x
n)n < (1+ t
n)n < ex+.
(1 + xn)n ex .
2.8 .. (xn) Cauchy > 0 n0 N
|xn xm| < n,m N, n,m n0 . :
limn,m+
(xn xm) = 0.
: (xn) Cauchy .
2.27. (xn) , Cauchy.
. xn x. > 0. n0 N |xn x| < 2 n N, n n0 . , n m, |xm x|
< 2 m N, m n0 .
|xn xm| = |(xn x) (xm x)| |xn x|+ |xm x| < 2 +2=
n,m N, n,m n0 . (xn) Cauchy.
2.3. (xn) Cauchy, .
. n0 N |xnxm| < 1 n,m N, n,m n0. , n N, n n0 |xn xn0 | < 1
, ,
|xn| = |(xn xn0) + xn0 | |xn xn0 |+ |xn0 | < 1 + |xn0 |.
M = max{|x1|, . . . , |xn01|, 1 + |xn0 |}.
|xn| M n N, 1 n n0 1 |xn| < 1+ |xn0 | M n N,n n0 . |xn| M n
N, (xn) .
Cauchy 2.27.
