n=1 n

∞

(iii) Suppose an ≥ 0 for each n ≥ 1. Show that n=1 an converges if and

N (j)

only if we can ¬nd a sequence N (j) ’ ∞ such that n=1 an tends to a limit

as j ’ ∞.

(iv) [This is an variation on the integral test described in Lemma 9.2.4.]

Suppose that we can ¬nd a positive continuous function f : [1, ∞] ’ R and

a constant K with K ≥ 1 such that

Kan ≥ f (t) ≥ K ’1 an ≥ 0 for all n ¤ t ¤ n + 1.

∞

Show that ∞ an converges if and only if 0 f (t) dt does.

n=1

(v) Find a sequence an ∈ R such that 2N an tends to a limit as N ’ ∞

n=1

but ∞ an does not converge.

n=1

(vi) Let an ∈ Rm . Suppose that there exists a strictly positive integer M

such that M N an tends to a limit as N ’ ∞ and suppose that an ’ 0 as

n=1

n ’ ∞. Show that ∞ an converges.

n=1

(vii) Find a sequence an ∈ R and a sequence N (j) ’ ∞ as j ’ ∞

N (j)

such that n=1 an tends to a limit as N ’ ∞ and an ’ 0 as n ’ ∞, yet

∞

n=1 an diverges.

Exercise K.69. [5.2, P] Let an , bn be sequences of non-negative real num-

bers.

4

Minkowski was walking through G¨ttingen when he passed a young man lost in deep

o

thought. ˜Don™t worry™ said Minkowski, ˜it is sure to converge.™

467

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(i) Show that, if ∞ an and ∞ bn converge, so does ∞ (an bn )1/2 .

n=1 n=1 n=1

(ii) Show that, if ∞ an converges, so does ∞ (an an+1 )1/2 .

n=1 n=1

(iii) Show that, if the an form a decreasing sequence, then, if ∞ (an an+1 )1/2

n=1

converges, so does ∞ an .

n=1

(iv) Give an example with ∞ (an an+1 )1/2 convergent and ∞ an di-

n=1 n=1

vergent.

Exercise K.70. [5.2, P] We work in the real numbers. Are the following

true or false? Give a proof or counterexample as appropriate.

(i) If ∞ a4 converges, then ∞ a5 converges.

n=1 n n=1 n

(ii) If n=1 an converges, then ∞ a4 converges.

∞ 5

n=1 n

∞

(iii) If an ≥ 0 for all n and n=1 an converges, then nan ’ 0 as n ’ ∞.

(iv) If an ≥ 0 for all n and ∞ an converges, then n(an ’ an’1 ) ’ 0 as

n=1

n ’ ∞.

(v) If an is a decreasing sequence of positive numbers and ∞ an con-

n=1

verges, then nan ’ 0 as n ’ ∞.

(vi) If an is a decreasing sequence of positive numbers and nan ’ 0 as

n ’ ∞, then ∞ an converges.

n=1

(vii) If n=1 an converges, then ∞ n’3/4 an converges.

∞

n=1

[Hint: Cauchy-Schwarz]

(ix) If ∞ an converges, then ∞ n’1 |an | converges.

n=1 n=1

Exercise K.71. [5.2, T] Show that if an ≥ 0 and k is a strictly positive

integer, the convergence of ∞ ak implies the convergence of ∞ ak+1 .

n=1 n n=1 n

Suppose now that we drop the condition that all the an be positive. By

considering a series of real numbers an with

a3n+1 = 2f (n), a3n+2 = ’f (n), a3n+3 = ’f (n)

for a suitable f (n) show that we may have

∞ ∞

ak divergent for all k ≥ 2.

an convergent but n

n=1 n=1

Exercise K.72. (Euler™s γ.) [5.2, T] Explain why

n+1

1 1

¤ dx.

n+1 x

n

Hence or otherwise show that if we write

n

1

’ log n

Tn =

r

r=1

468 A COMPANION TO ANALYSIS

we have Tn+1 ¤ Tn for all n ≥ 1. Show also that 1 ≥ Tn ≥ 0. Deduce that

Tn tends to a limit γ (Euler™s constant) with 1 ≥ γ ≥ 0. [It is an indication

of how little we know about speci¬c real numbers that, after three centuries,

we still do not know whether γ is irrational. Hardy is said to have o¬ered

his chair to anyone who could decide this.]

(ii) By considering T2n ’ Tn , show that

1111111

1’ + ’ + ’ + ’ · · · = log 2

2345678

(iii) By considering T4n ’ 2 T2n ’ 1 Tn , show that

1

2

11111 3

’ + + ’ + · · · = log 2

1+

32574 2

This famous example is due to Dirichlet. It gives a speci¬c example where

rearranging a non-absolutely convergent sum changes its value.

(iv) Show that

11 1 1 1

+ + ··· + ’ log n ’ γ,

24 2n 2 2

11 1 1 1

1 + + + ··· + ’ log n ’ γ + log 2

35 2n + 1 2 2

as n ’ ∞.

Deduce that

1 1 11 1 1 1 1

’ ’ ’ ··· ’ + ··· + ’ ...,

1+ + ... + +

2p ’ 1 2 4 4p ’ 1

3 2q 2p 2p + 2

where p positive terms alternate with q negative terms, has sum log 2 +

(1/2) log(p/q).

(v) Use the ideas of (iv) to show how the series may be arranged to

converge to any desired sum. (Be careful, convergence of a subsequence of

sums does not imply convergence of the whole sequence.)

Exercise K.73. [5.3, P] We work in C.

(i) Review the proof of Abel™s lemma in Exercise 5.2.6. Show that, if

| N aj | ¤ K for all N and »j is a decreasing sequence of positive terms

j=0

with »j ’ 0 as j ’ ∞, then ∞ »j aj converges and

j=0

∞

»j aj ¤ K sup »j .

j≥0

j=0

469

Please send corrections however trivial to twk@dpmms.cam.ac.uk

∞

(ii) Suppose that j=0 bj converges. Show that, given > 0, we can ¬nd

an N ( ) such that

∞

bj xj ¤

j=M

whenever M ≥ N ( ) and x is a real number with 0 ¤ x < 1. (This result

was also obtained in Exercise K.61 (vii) but the suggested proof ran along

di¬erent lines. It worth mastering both proofs.)

(iii) By considering the equation

∞ M ’1 ∞

bj xj = bj xj + bj xj ,

j=0 j=0 j=M

∞

or otherwise, show that, if j=0 bj converges, then

∞ ∞

bj xj ’ bj

j=0 j=0

when x ’ 1 through real values of x with x < 1.

(iv) Use the result of Exercise 5.4.4 together with part (iii) to prove the

following result. Let aj and bj be sequences of complex numbers and write

n

cn = an’j bj .

j=0

∞ ∞ ∞

Then, if all the three sums aj , j=0 bj and j=0 cj converge (not nec-

j=0

essarily absolutely), we have

∞ ∞ ∞

aj bj = cj .

j=0 j=0 j=0

(v) By choosing aj = bj = (’1)j j ’1/2 , or otherwise, show that, if cn is

de¬ned as in (iv), the two sums ∞ aj and ∞ bj may converge and yet

j=0 j=0

∞

j=0 cj diverge.

Exercise K.74. [5.3, P] This exercise is fairly easy but is included to show

that the simple picture painted in Theorem 4.6.19 of a ˜disc of convergence™

for power series fails in more general contexts. More speci¬cally we shall

deal with the absolute convergence of ∞ ∞

with cn,m ∈ R

nm

m=0 cn,m x y

n=0

[n, m ≥ 0] and x, y ∈ R. (We shall deal with absolute convergence only

470 A COMPANION TO ANALYSIS

because as we saw in Section 5.3 this means that the order in which we sum

terms does not matter.)

(a) Show that, if there exists a δ > 0 such that ∞ ∞ nm

m=0 cn,m x y

n=0

converges absolutely for |x|, |y| ¤ δ then there exists a ρ > 0 and an K > 0

such that

|cn,m | ¤ Kρn+m for all n, m ≥ 0.

(b) Identify the set

∞ ∞

E = {x, y ∈ R :2

|cn,m xn y m | converges}

n=0 m=0

in the following cases.

(i) cn,m = 1 for all n, m ≥ 0.

(ii) cn,m = n+m for all n, m ≥ 0.

n

(iii) c2n,2m = n+m for all n, m ≥ 0 and cn,m = 0 otherwise.

n

(iv) cn,m = (n!m!)’1 for all n, m ≥ 0.

(v) cn,m = n!m! for all n, m ≥ 0.

(vi) cn,n = 1 for all n ≥ 0. and cn,m = 0 otherwise.

(c) Let E be as in (i). Show that if (x0 , y0 ) ∈ E then (x, y) ∈ E whenever

|x| ¤ |x0 | and |y| ¤ |y0 |. Conclude that E is the union of rectangles of the

form [’u, u] — [’v, v].

Exercise K.75. [5.3, T!] Part (ii) of this fairly easy question proves a ver-

sion the so called ˜Monotone Convergence Theorem™. Suppose that aj,n ∈ R

for all j, n ≥ 1, that aj,n+1 ≥ aj,n for all j, n ≥ 1, and aj,n ’ aj as n ’ ∞

for all j ≥ 1 .

(i) Show that, given any > 0 and any M ≥ 1, we can ¬nd an N ( , M )

such that

∞ M

aj,n ≥ aj ’

j=1 j=1

for all n ≥ N ( , M ). Show also, that, if ∞ aj converges, we have ∞ aj ≥

j=1 j=1

∞

aj,n for all n.

j=1

∞

(ii) Deduce that, under the hypotheses of this exercise, j=1 aj,n con-

∞ ∞

aj,n ’ A as n ’ ∞ if and only if

verges for each n and j=1 aj

j=1

converges with value A.

Exercise K.76. [5.3, T!, ‘ ] In this question we work in C. If ∞ an z n

n=0

∞ n

converges to f (z), say, for all |z| < δ and n=0 bn z converges to g(z), say,

471

Please send corrections however trivial to twk@dpmms.cam.ac.uk

for all |z| < δ [δ, δ > 0] it is natural to ask if f (g(z)) = f —¦ g(z) can be

written as a power series in z for some values of z.

(i) Show that formal manipulation suggests that

∞ ∞

?

cn z n with cn =

f —¦ g(z) = ar bm(1) bm(2) . . . bm(r) .

n=0 r=0 m(1)+m(2)+···+m(r)=n

The rest of this question is devoted to showing that this equality is indeed

true provided that

∞

|bn z n | < δ

n=0

where δ is the radius of convergence of ∞ an z n . The exercise is only worth

n=0

doing if you treat it as an exercise in rigour. Since the case z = 0 is fairly

trivial we shall assume that z = 0.

(ii) Let us de¬ne cN r and CN r by equating coe¬cients of powers of w in

the equations

n n

∞ ∞

N N N N

cN r w r = bm w m CN r w r = |bm |wm

|an |

an and .

r=0 n=0 m=0 r=0 n=0 m=0

(Note that cN r = CN r = 0 if r is large.)

Show that CN q |z|r ¤ ∞ |an |κq where κ = ∞ |bn z n |. By using the

n=0 n=0

fact that an increasing sequence bounded above converges show that CN q ’

Cq for some Cr as N ’ ∞. Use the monotone convergence theorem (Ex-

ercise K.75) to show that ∞ |an | m(1)+m(2)+···+m(r)=n |bm(1) bm(2) . . . bm(r) |

n=0

converges to Cn . Deduce that ∞ ar m(1)+m(2)+···+m(r)=n bm(1) bm(2) . . . bm(r)

r=0

converges and now use the dominated convergence theorem (Lemma 5.3.3)

to show that

∞

cN n ’ ar bm(1) bm(2) . . . bm(r) .

r=0 m(1)+m(2)+···+m(r)=n

(iii) By further applications of the monotone and dominated convergence

theorems to both sides of the equations

n n

∞ ∞

N N N N

CN r |z|r = |bm z|m cN r z r = bm z m

|an | and an

r=0 n=0 m=0 r=0 n=0 m=0

472 A COMPANION TO ANALYSIS

show that the two sides of the following equation converge and are equal.

n

∞ ∞

N

m

cn z n

an bm w =

n=0 m=0 n=0

where cn = ∞ br m(1)+m(2)+···+m(r)=n bm(1) bm(2) . . . bm(r) .

r=0

(iv) Let us say that a function G : C ’ C is ˜locally expandable in power

series at z0 ™ if we can ¬nd an · > 0 and Aj ∈ C such that ∞ Aj z j has

j=0

∞

radius of convergence at least · and G(z) = j=0 Aj (z ’ z0 ) for all z ∈ C

j

with |z ’ z0 | < ·. Show that if G : C ’ C is locally expandable in power

series at z0 and F : C ’ C is locally expandable in power series at G(z0 ) then

F —¦ G is locally expandable in power series at z0 . [In more advanced work

this result is usually proved by a much more indirect route which reduces it

to a trivial consequence of other theorems.]

Exercise K.77. (The Wallis formula.) [5.4, P] Set

π/2

sinn x dx.

In =

0

n

In’1 for n ≥ 0.

(i) Show that In+1 =

n+1

(ii) Show that In+1 ¤ In ¤ In’1 and deduce that In+1 /In ’ 1 as n ’ ∞.

(iii) By computing I0 and I1 directly, ¬nd I2n+1 and I2n for all n ≥ 0.

(iv) By applying (ii) and (iii), show that

n

4k 2 π

’ as n ’ ∞.

4k 2 ’ 1 2

k=1

Exercise K.78. (In¬nite products.) [5.4, T]

(i) Prove that, if x ≥ 0, then exp x ≥ 1 + x.

(ii) Suppose a1 , a2 , . . . , an ∈ C. Show that

n n

(1 + aj ) ¤ (1 + |aj |)

j=1 j=1

and

n n

(1 + aj ) ’ 1 ¤ (1 + |aj |) ’ 1.

j=1 j=1

473

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iii) Suppose a1 , a2 , . . . , an , . . . ∈ C and 1 ¤ n ¤ m. Show that

n m n m

(1 + aj ) ’ (1 + aj ) ¤ exp |aj | |aj | ’1 .

exp

j=1 j=1 j=1 j=n+1

(iv) Show that, if ∞ aj converges absolutely, then n

j=1 (1 + aj ) tends

j=1

to a limit A, say. We write

∞

A= (1 + aj ).

j=1

(v) Suppose that ∞ aj converges absolutely, and, in addition that an =

j=1