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Pure and Applied Analysis Problems, 1st Semester 2010-2011 1

Pure and Applied Analysis Tutorial Problems. The problems are of varying difficulty.The E questions are fairly routine exercises in the coursework, S questions are of standard difficultyand H questions may be quite challenging. P questions are more peripheral to the course.

You are encouraged to hand in solutions to problems 1 and 14 in week 3.

S1. In each of the following cases a bounded subset A ofR is given. Find max A and min A (if

they exist) as well as sup Aand infA. Prove your claims.(i) A={(1)n :n N}(ii) A={ 1

n :n N}

(iii) A={x R :x is rational and 0x 0, (b) n

n, (c) n

n!, (d) n!nn

, (e) 100n

n! , (f)sin

n2

,

(g)

n2 + 1 n, (h) n2 +n n.S11. Show that

1 + 1

n

n e as n .E12. What is the limit of

1 + 1

n2

n?

E13. A sequence an has the property,

nN, an+1= 2an

3

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Show that(i) Ifa1= 3 then for all nN, an = 3.(ii) Ifa1 >3 then an+.(iii) Ifa1 1, (c) n

3n

n! , (d)n

cos n

n2 ,

(e)n

1

log n, (f)

n

(1)nn

, (g)n

n n 1, (h)

n=3

1

n log n log log n, (i)

n

sinn

2 .

S25. Let , (0, ) and consider the seriesn=2 1n(logn) . Show that:(i) If >1 then the series converges.(ii) If= 1 and >1 then the series converges.(iii) In all other cases the series diverges.

E26. Let n

an be a convergent series of positive terms. Show that n

a2n converges.

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You are encouraged to hand in a solution to problem 31 in week 4, 46 in week 5 and 50 in week 6.

S27. Let (an) be a sequence of real numbers. If the series

na2n converges, show that

nann

converges. Is the converse true? [Hint: Use the Cauchy-Schwarz inequality.]

E28. Let (an) be a decreasing sequence converging to zero. Show that for every N Nwe have(i) a2N

a2N+1+

0.

(ii)a2N+1+a2N+2 0P29. If (an) is a sequence of positive real numbers numbers with limn

an+1an

= l R thenlimn n

an = l.

S30. Define a sequence (an) bya2n= 1 , n= 1, 2, 3, . . .and a2n+1 = 2n , n= 0, 1, 2, 3, . Show

that n

an1 but an+1an has no limit.S31. Let (an) be a sequence of real numbers such that an0 as n . Prove that there existsa subsequence (ank) such that

k

ank converges. [Hint: Construct a subsequence (ank) such that

for all k,

|ank

| 1k2

.]

S32. Prove that the sequence an=n1

k=11k log n (n N) converges. [Hint: It is monotone and

bounded.]

S33. Find the radius of convergence of the following power series.

(i)n

nn

n!zn (ii)

n

nzn (iii)n

zn

n!

(iv)n

(log n)zn (v)n

n

4n +nzn.

E34. Can a power series of the form n

an(z 2)n converge at z= 0 but diverge at z= 2 +i?

S35. Let (an) and bn be two sequences of complex numbers. For nNdefine Bn =b1+ +bn,and set B0= 0. Prove that for any positive integers N , M with N > M,

Nn=M

anbn=N

n=M

an(Bn Bn1) =aNBN aMBM1 N1n=M

(an+1 an)Bn

H36. Consider the power series

n

nzn , n

zn

n2 ,

n

zn

n

Show that in all three cases the radius of convergence is R = 1. Show that the first converges at nopoint on the unit circle, the second converges at all points on the unit circle and the third convergesat all points on the unit circle except z= 1.

S37. Suppose that the power series

nanzn has radius of convergence R (0, ) and that all

an are nonnegative real numbers. Prove that if the series converges at z = R then it convergesabsolutely at all points on the circle|z|= R.P38. Show that if all an are positive integers then the radius of convergence of the power series

nanzn is at most one.

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P39. Let (an) be a sequence of complex numbers. Show that the power series

anz

n ,

nanzn ,

an

zn+1

n+ 1

have the same radius of convergence.

S40. Let fn : RR(nN) be a sequence of continuous functions which converges uniformly toa function f : R R. Let (xn) be a sequence of real numbers which converges to a real number x.Show that fn(xn)f(x).S41. Letfn: R R (n N) be a sequence of continuous functions. Letf : R R be a continuousfunction. Suppose that for any sequence (xn) of real numbers which converges to any real limit x,the sequence (fn(xn)) converges to f(x).(i) Does it follow that fnf uniformly?(ii) Does it follow that fnf pointwise?S42. Prove that the sequence of functions fn : RR , fn(x) = x

n

1+nx2converges pointwise to the

zero function. Is the convergence uniform over R?

S43. Consider the sequence (fn) of functions defined on [0, 1) by fn(x) =nxn. Show that fn 0

pointwise but10

fn(x)dx1.S44. Consider the sequence of functions on R given by fn(x) = (x 1n)2. Prove that it convergespointwise and find the limit function. Is the convergence uniform on R? Is the convergence uniformon bounded intervals?

S45. Let fn(x) = x xn. Prove that fn converges pointwise on [0, 1] and find the limit function.Is the convergence uniform on [0, 1] ? Is the convergence uniform on [0, 1)?

S46. Consider the sequence of functions defined on [0, ) byfn(x) = xn1+xn .(i) Prove that (fn) converges pointwise and find the limit function.(ii) Is the convergence uniform on [0, )?(iii) Is the convergence uniform on bounded intervals of the form [0 , a)?

S47. Let fn(x) = nx1+n2x2

.(i) Prove that (fn) converges pointwise on Rand find the limit function.(ii) Is the convergence uniform on [0, 1]?(iii) Is the convergence uniform on [1, )?S48. Define fn on [0, ) byfn(x) = nx1+nx(i) Prove that (fn) converges pointwise and find the limit function.(ii) Is the convergence uniform on [0, 1]?(iii) Is the convergence uniform on (0, 1]?(iv) Is the convergence uniform on [1, )?

E49. Find limn

10

2n+ sin x

3n+ cos2 xdx.

S50.(i) Prove that the sequence of functions fn(x) =nx(1 x2)n converges pointwise on [0, 1] andfind the limit function. (You may use without proof that: nan 0 for a(0, 1)).(ii) Is the convergence uniform on [0, 1]? (Hint: Consider the integrals

10

fn.)(iii) Is the convergence uniform on [a, 1] where 0< a

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You are encouraged to hand in a solution to problem 59 in week 7 and problem 70 in week 8.

E52. Let fn(x) =

x2 +n2. Show that the sequence converges uniformly over R and find thelimit function f.

Show that each fn is differentiable but that fis not differentiable.

E53. Consider the sequence of functions fn : R

R given by fn(x) = n

2 sin(n2x). Show that

the sequence converges uniformly on R and find the limit function f. Show also that each fn isdifferentiable. Show that the sequence of the derivatives (fn) does not converge (even pointwise)on R.

We have fn() = cos(n2) = (1)n, which does not converge, sofn does not converge pointwise on

R.

S54. Find the subset of (0, ) on which the sequence of functions fn(x) = n| log x|n convergespointwise. Is the convergence uniform?

H55. Let R. Find the largest subset of R on which the sequence of functions fn(x) =xn

enx logn (n2) converges pointwise. Is the convergence uniform on that set?

S56. Let f : [1, 1] R be a continuous function. Suppose that there is an L (0, 1) suchthat for all x [1, 1],|f(x)| L|x| Define a sequence of functions on [1, 1] by f1 = f andfn(x) =f(fn1(x)). Show that this sequence converges uniformly to the zero function.

H57. Let fn : [a, b] R be a sequence of continuous functions which converges uniformly onthe open interval (a, b) to a function f : (a, b) R. Show that the sequences (fn(a)) and (fn(b))converge. Call their limits A and B respectively. Then show that the sequence of functions (fn)converges uniformly on the closedinterval [a, b] to the function Fdefined on [a, b] byF(x) =f(x)for x(a, b) and F(a) =A, F(b) =B .H58. (Dinis Theorem) Let (fn) be a decreasing sequence of continuous functions on the interval

[a, b] which converges pointwise to zero. Show that (fn) converges uniformly to zero.S59. Show that the series of functions

n=1ne

nx converges uniformly on [1, ) to a (continuous)function f. [Hint: Use the Weierstrass M-test]. Find

21

f(x)dx.

S60. Consider the series of functions

n=021nxn forx(2, 2). Prove that it converges pointwise

and find the limit function. Is the convergence uniform on (2, 2)? Let(0, 2). Is the convergenceuniform on (, )?

S61. Consider the series of functionsn=0

xn

n!, which converges pointwise to the exponential function.

Show that the convergence is uniform on every bounded interval. Is the convergence uniform on R?

S62. Is the function fdefined on [0, 1] by

f(x) =

x, if x is rational0 if x is irrational

Riemann integrable?

E63. Give an example of a function f : [0, 1] R which is not Riemann integrable for which|f|is Riemann integrable.

S64. Let g : [a, b]

Rbe continuous and non-negative. Ifba

g(x)dx= 0, show that g = 0 on [a, b].

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S65. Let f , g: [a, b]Rbe continuous and suppose that g0. Show that there exists a [a, b]such that b

a

f(x)g(x)dx= f()

ba

g(x)dx

S66. Let f : [0, a][0, b] be continuous, one-to-one and onto. Explain why a

0

f(x)dx+ b

0

f1(y)dy = ab

[Hint: There is a very simple geometric reason for this result]

S67. Let f, g: [a, b] R be Riemann integrable. Prove the Cauchy-Schwarz inequality: ba

f(x)g(x)dx b

a

f(x)2dx

12 b

a

g(x)2dx

12

[Hint: ba (f(x) g(x))2 dx

0 for any constant

R]

S68. Let f : R Rbe locally Riemann integrable. Show that the function

g(x) =

x0

f(t)dt

is continuous.

H69. Let f : [a, b] Rbe continuous and let M = maxx[a,b]

|f(x)|. Show that

limn

b

a |f(x)

|ndx

1/n

=M

S70. Show that the function f(x) =|x|1/2 is uniformly continuous on R. (Hint: when estimatingf(x) f(y), you may find it helpful to consider separately the case when xand y are both close to0).

S71. Supposefis a bounded function on a closed interval [a, b], and that fis continuous on (a, b).Show that f is Riemann integrable on [a, b], without using Theorem 2 of section 3.4.

Deduce that the function fdefined on [0, 1] by

f(x) = sin 1

x, 0< x1

0, x= 0

is Riemann integrable on [0, 1].

E72. Show that for any two integers n, m the following orthogonality relations hold:

(i) 1

2

20

cos nx cos mxdx= 0 ifn=m and 12

ifn= m

(ii) 1

2

20

sin nx sin mx dx= 0 ifn=m and 12

ifn= m

(i11) 1

2

20

sin nx cos mx dx= 0.

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You are encouraged to hand in a solution to problem 73 in week 9, 83(ii) in week 10 and 90 inweek 11.

S73. Fix x R. Show that:(i) The sequence (einx) converges if and only ifx2Z. (Hint: consider einx ei(n+1)x).(ii) The sequence (sin(nx)) converges if and only ifx

Z.

(iii) The sequence (cos(nx)) converges if and only ifx2Z.S74. Let fbe defined on [, ) byf(x) =

x, x

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S84. Let f : R Cbe periodic. Fix an integer m. Show that:1

2

einxf(mx)dx=

fnm

, ifndivides m

0, ifndoesnt divide m

S85. Let f : R C be periodic. Fix an integerm and define g(x) = f(mx). Iff(x) has Fourierseries

n= cneinx show that f(mx) has Fourier seriesn= cneinmx.S86. Find the Fourier series of the periodic function determined by f(x) = x, < x andthen use Parsevals identity to show that

n=1

1n2

= 2

6.

S87. Show that there is a unique sequence of polynomials P0, P1, such that P0(x) = 1, andPn(x) =Pn1(x) and

Pn(x)dx= 0 for n1.

Verify that P1(x) =x and P2(x) = x2

2 2

6, and find P3(x).

Now let cn,k = 12

e

ikxPn(x)dx, the complex Fourier coefficients of Pn. Show that cn,k =1ik

cn1,k for n1 and k= 0. (Hiny: first show that Pn() =Pn() for n >1).Calculate c1,k and deduce that cn,k = (1)

k

(ik)n for n1 and k= 0.

Then use Parsevals relation to show thatk=1

k2n = 1

4

P2n

for n= 1, 2, . Hence show thatk=1 1k6 = 6945 .S88. Let a, b R with a= 0, and define f on [, ) by f(t) = e(a+ib)t. Calculate the complexFourier coefficients ck off. By applying Parsevals formula, show that

k=

1

a2 + (b k)2 =

sinh2a

a(cosh2a cos2b)S89. Let a, bRwith a= 0, and define f on [, ) byf(t) = (x b)eax. Calculate the complexFourier coefficients ck off.

Now suppose a= 0 is given. By choosing an appropriate b and applying Parsevals formula,show that

k=

1

(a2 +k2)2 =

{a(cosech a)2 + coth a}2a3

S90. Suppose that fhas a continuous derivative on [, ] and that f() = f(). Let ck =12 e

ikxf(x)dxand dk = 12 eikxf(x)dx, the complex Fourier coefficients offandf. Show

thatdk =ikckand deduce that

k= k2|ck|2

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