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fn (yn )
r=0
555
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Deduce the existence of an in¬nitely di¬erentiable function f : R ’ R
whose Taylor series

f (n) (q)
(q + h)n
n!
n=0

diverges for all h = 0 and all rational numbers q.
(iii) The result of (ii) is even stronger than at ¬rst appears. Use the ideas
of Exercise K.230 to show that, if a function F : R ’ R has derivatives of
all orders at x and

F (n) (x) n
F (x + h) = h
n!
n=0

for all |h| < R, then F is in¬nitely di¬erentiable at t for all |t ’ x| < R and

F (n) (t) n
F (t + k) = k
n!
n=0

whenever |k| < R ’ |t ’ x|.
Conclude that the function f obtained in (ii) can not have a valid Taylor
expansion about any point.
Exercise K.227. [11.4, T!] The object of this question and the next two is
to de¬ne x± and to obtain its properties directly rather than by the indirect
de¬nition used in Section 5.7. In this ¬rst question we deal with the case ±
rational. We assume the properties of xn when n is an integer.
(i) Carefully quoting the results that you use, show that, if n is a strictly
positive integer, then, given any x > 0, there exists a unique r1/n (x) > 0 with
r1/n (x)n = x. Show further (again carefully quoting the results that you use)
that the function r1/n : (0, ∞) ’ (0, ∞) is di¬erentiable with

r1/n (x)
r1/n (x) = .
nx
(ii) Show that if m, m , n and n are strictly positive integers with m/n =
m /n then

(r1/n (x))m = (r1/n (x))m

for all x > 0. Explain why this means that we can de¬ne rm/n (x) = r1/n (x)m
for all m, n positive integers.
(iii) If ±, β ∈ Q and ±, β > 0 prove the following results.
556 A COMPANION TO ANALYSIS

(a) r±+β (x) = r± (x)rβ (x) for all x > 0.
(b) r±β (x) = r± (rβ (x)) for all x > 0.
(c) r± (xy) = r± (x)r± (y) for all x, y > 0.
(d) r± (1) = 1.
(e) The function r± : (0, ∞) ’ (0, ∞) is di¬erentiable with
±r± (x)
r± (x) = .
x
Show further that, if n is a strictly positive integer,
(f) rn (x) = xn for all x > 0.
(iv) Show carefully how to de¬ne r± for all ± ∈ Q and obtain results
corresponding to those in part (iii).
Exercise K.228. [11.4, T!, ‘ ] We continue the arguments of Question K.227.
(i) Suppose that x > 0, ± ∈ R and ±n ∈ Q with ±n ’ ± as n ’ ∞.
Show that the sequence r±n (x) is Cauchy and so tends to a limit y, say.
Suppose that βn ∈ Q with βn ’ ± as n ’ ∞. Show that rβn (x) ’ y as
n ’ ∞. We can thus write r± (x) = y.
Show also that r± (x) > 0.
(ii) Suppose that ± ∈ R, ± = 0 and ±n ∈ Q with ±n ’ ± as n ’ ∞.
Show that, if 0 < a < b, we have r±n ’ r± uniformly on [a, b]. Deduce,
quoting any results that you use, that the function r± : (0, ∞) ’ (0, ∞) is
di¬erentiable with
±r± (x)
r± (x) = .
x
(iii) Prove the remaining results corresponding to to those in part (iii) of
Question K.227.
Exercise K.229. [11.4, T!, ‘ ] Question K.228 does not give all the prop-
erties of x± that we are interested in. In particular, it tells us little about
the map ± ’ x± . If a > 1, let us write P (t) = Pa (t) = at .
(i) Show that P : R ’ (0, ∞) is an increasing function.
(ii) Show that P (t) ’ 1 as t ’ 0.
(iii) By using the relation P (s+t) = P (s)P (t), show that P is everywhere
continuous.
(iv) Show that, if P is di¬erentiable at 1, then P is everywhere di¬eren-
tiable.
(v) Let f (y) = n(y 1/n ’ 1) ’ (n + 1)(y 1/(n+1) ’ 1). By considering f , or
otherwise, show that f (y) ≥ 0 for all y ≥ 1. Hence, or otherwise, show that
P (1/n) ’ P (0)
tends to a limit, L say,
1/n
557
Please send corrections however trivial to twk@dpmms.cam.ac.uk

as n ’ ∞.
(vi) Show that, if 1/n ≥ x ≥ 1/(n + 1) then

P (x) ’ P (0) P (1/n) ’ P (0) n + 1
¤ — .
x 1/n n
By using this and similar estimates, or otherwise, show that
P (x) ’ P (0)
’L
x
as x ’ 0 through values of x > 0.
(vii) Show that, if x = 0

P (x) ’ P (0) P (’x) ’ P (0)
= P (x)
’x
x
and deduce that
P (x) ’ P (0)
’L
x
as x ’ 0. Conclude that P is di¬erentiable at 0 and so everywhere with
P (t) = LP (t).
(viii) If a > 0 and we set Pa (t) = at , show that Pa is di¬erentiable with
Pa (t) = La P (t) for some constant La . Show that La > 0 if a > 1. What can
you say about La for other values of a?
(ix) If a, » > 0 show that La» = »La . Deduce that there is a real number
e > 1 such that, if we write

e(t) = et ,

then e is a di¬erentiable function with e (t) = e(t).
(x) Tear o¬ the disguise of La .
We have now completed a direct attack on the problem of de¬ning x± and
obtaining its main properties. It should be clear why the indirect de¬nition
x± = exp(± log x) is preferred.
Exercise K.230. [11.5, P] Suppose ∞ an z n has radius of convergence
n=0
greater than or equal to R > 0. Write g(z) = ∞ an z n for |z| < R. Show
n=0

that, if |z0 | < R then we can ¬nd b0 , b1 , · · · ∈ C such that n
n=0 bn z
has radius of convergence greater than or equal to R ’ |z0 | and g(z) =
∞ n
n=0 bn (z ’ z0 ) for |z0 ’ z| < R ’ |z0 |. [This is quite hard work. If you
have no idea how to attack it, ¬rst work out formally what the values of the
bn must be. Now try and use Lemma 5.3.4.]
558 A COMPANION TO ANALYSIS

Show that g is complex di¬erentiable at z0 with g (z0 ) = b1 = ∞ nan z0 .n’1
n=1
Deduce that a power series is di¬erentiable and that its derivative is that ob-
tained by term by term di¬erentiation within its radius of convergence. (We
thus have an alternative proof of Theorem 11.5.11.)

Exercise K.231. [11.5, P] Here is another proof of Theorem 11.5.11. Sup-
pose ∞ an z n has radius of convergence greater than or equal to R > 0.
n=0
(i) Show that ∞ nan z n’1 and ∞ n(n ’ 1)an z n’2 also have radius
n=1 n=2
of convergence R.
(ii) Show that

n’2
n
¤ n(n ’ 1)
j j

for all n ’ 2 ≥ j ≥ 0 and deduce that

|(z + h)n ’ z n ’ nz n’1 h| ¤ n(n ’ 1)(|z| + |h|)n’2

for all z, h ∈ C.
(iii) Use (i) and (ii) to show that, if 0 < δ and |z| + |h| < R ’ δ, then
∞ ∞ ∞
n n
nan z n’1 ¤ A(δ)|h|2
an (z + h) ’ an z ’ h
n=0 n=0 n=1

where A(δ) depends only on δ (and not on h or z).
(iv) Deduce that a power series is di¬erentiable and that its derivative is
that obtained by term by term di¬erentiation within its radius of convergence.


Exercise K.232. [11.5, P] Consider the Legendre di¬erential equation

(1 ’ x2 )y ’ 2xy + l(l + 1)y = 0,

where l is a real number. Find the general power series solution. Show that,
unless l is non-negative integer, every non-trivial power series solution has
radius of convergence 1.
Show, however, that, if l = n a non-negative integer the general series
solution can be written

y(x) = APn (x) + BQn (x)

where Qn is a power series of radius of convergence 1, Pn is a polynomial of
degree n, and A and B are arbitrary constants.
559
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Verify that, when n is a non-negative integer, the function given by v(x) =
(x2 ’ 1)n satis¬es the equation

(1 ’ x2 )v (x) + 2nxv(x) = 0.

Deduce that v (n) (x) satis¬es the Legendre di¬erential equation with l = n.
[Consult Exercise K.270 if you need a hint.] Conclude that Pn is a constant
multiple of the function pn de¬ned in Exercise K.212.

Exercise K.233. [11.5, P] Obtain the general power series solution of
2
2d w dw
+ (z 2 ’ 1)w = 0.
z +z
dz 2 dz
For what values of z is your solution valid and why?
Answer the same questions for
2
2d w dw
+ (z 2 ’ 1)w = z 2 .
z +z
dz 2 dz
Exercise K.234. [11.5, P] Using Lemma 11.5.19 or otherwise, ¬nd the
power series expansion ∞ aj xj of (1 + x)1/2 for x real and |x| < 1.
j=0
Show that the complex power series ∞ aj z j has radius of convergence
j=0
1. What is the power series for the product ∞ aj z j ∞ aj z j and why?
j=0 j=0
What is its radius of convergence?
By considering
∞ ∞ ∞
aj K ’j z j
aj z j aj z j ,
j=0 j=0 j=0


or otherwise, show that, if two power series of the same radius of conver-
gence R are multiplied, the resulting power series may have any radius of
convergence with value R or greater.
By considering expressions of the form
∞ ∞ ∞
Aj z j Bj z j + Cj z j ,
j=0 j=0 j=0


or otherwise, show that, if two power series of radius of convergence R and S
are multiplied, the resulting power series may have any radius of convergence
with value min(R, S) or greater.
560 A COMPANION TO ANALYSIS

Exercise K.235. [11.5, P] By modifying the proof of Abel™s test (Lemma 5.2.4),
or otherwise, prove the following result. Let E be a set and aj : E ’ Rm a
sequence of functions. Suppose that there exists a K such that
n
aj (t) ¤ K
j=1

for all n ≥ 1 and all t ∈ E. If »j is a decreasing sequence of real positive
numbers with »j ’ 0 as j ’ ∞ then ∞ »j aj converges uniformly on E.
j=1
Deduce, in particular, that if bn ∈ R and ∞ bn converges, then ∞ bn xn
n=0 n=0
converges uniformly on [0, 1]. Explain why this implies that
∞ ∞
1
bn
n
bn x dx = .
n+1
0 n=0 n=0

Show that, provided we interpret the left hand integral as an improper
1’
integral, lim ’0, >0 0 , equation remains true under the assumption

that n=0 bn /(n + 1) converges.
Show that
111
1 ’ + ’ + · · · = log 2
234
and
111 π
1 ’ + ’ + ··· = .
357 4
Do these provide good methods for computing log 2 and π? Why?
Recall that, if a, b > 0, then aloga b = b. Show that
log2 e ’ log4 e + log8 e ’ log16 e + . . .
converges to a value to be found.
Exercise K.236. [11.5, G, S] (Only if you have done su¬cient probability,
but easy if you have.) Let X be a random variable taking positive integral
values. Show that

φX (t) = Et = X
Pr(X = n)tn
n=0

is well de¬ned for all t ∈ [’1, 1].
If EX is bounded, show that
φX (1) ’ φX (t)
’ EX
1’t
as t ’ 1 through values t < 1. Give an example of an X with EX in¬nite.
561
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Exercise K.237. [11.5, T, S] Use the ideas of Exercise K.78 to prove the
following extension. Suppose that we have a sequence of functions an : „¦ ’ C
and a sequence of positive real numbers Mn such that ∞ Mj converges
j=1
absolutely and |an (z)| ¤ Mn for all z ∈ „¦ and all n. Then

N
(1 + aj (z)) ’ (1 + aj (z))
j=1 j=1


uniformly on „¦.

Exercise K.238. [11.5, P, S, ‘ ] We do not have the apparatus to exploit
Exercise K.237 fully but the following exercise is suggestive.
(i) Show that SN (z) = z N (1 ’ z 2 n’2 ) converges uniformly to S(z) =
n=1
z ∞ (1 ’ z 2 n’2 ) on any disc {z : |z| < R}. Deduce that S is a well de¬ned
n=1
continuous function on C.
(ii) By writing SN (z) = ’(N !)’2 N n=’N (n ’ z) and considering SN (z +
1)/SN (z), or otherwise, show that S(z + 1) = S(z) (that is, S is periodic
with period 1). Show also that S(z) = 0 if and only if z is an integer.
[The product z ∞ (1 ’ z 2 n’2 ) goes back to Euler who identi¬ed it with
n=1
a well known function which the reader should also be able to guess. An
advantage of in¬nite products over in¬nite sums is that we can ¬nd zeros of
the function much more easily.]

Exercise K.239. [11.5, T] (i) Suppose that an ∈ C and ∞ an z n has
n=0
radius of convergence R > 0. If aN = 0 and an = 0 for n < N , use the
inequality
q q
n N
|an z n’N |
¤ |z| |aN | ’
an z
n=0 n=N +1


(for q > n), together with standard bounds on |an z n |, to show that there
exists a δ > 0 such that
q
|aN ||z|N
n

an z
2
n=0

for all |z| < δ.
(ii) Suppose that an ∈ C and ∞ an z n has radius of convergence R > 0.
n=0
∞ n
Set f (z) = n=0 an z for |z| < R. Show, using (i), or otherwise, that, if
there exist wm ’ 0 with f (wm ) = 0 and wm = 0, then an = 0 for all n and
f = 0.
562 A COMPANION TO ANALYSIS

Exercise K.240. [11.5, P, ‘ ] This is a commentary on the previous exer-
cise. We shall consider functions from R to R.
(i) Suppose that an ∈ R and ∞ an xn has radius of convergence R > 0.
n=0
Set f (x) = ∞ an xn for |x| < R. Show that, if there exist xm ’ 0 with
n=0
f (xm ) = 0 and xm = 0, then an = 0 for all n and f = 0.
(ii) Explain why a polynomial of degree at most n with n + 1 zeros must
be zero.
(iii) Give an example of a power series of radius of convergence ∞ which
has in¬nitely many zeros but is not identically zero.
(iv) Let F be as in Example 7.1.5. Set G(x) = F (x) sin x’1 for x = 0
and G(0) = 0. Show that G is in¬nitely di¬erentiable everywhere (this will
probably require you to go through much the same argument as we used for
F ). Show that there exist xm ’ 0 with G(xm ) = 0 and xm = 0 but G is not
identically zero.
Exercise K.241. [11.5, P] This question prepares the way for Exercise K.242.
The ¬rst two parts will probably be familiar but are intended to provide a
hint for the third part.
n n n+1
(i) If n ≥ r ≥ 1, show that + = .
r’1 r r
(ii) Use induction to show that
n
n r n’r
(x + y)n = xy
r
r=0

whenever n is a positive integer and x and y are real.
(iii) Suppose that n is a positive integer and x and y are real. Verify the
next formula directly for n = 0, 1, 2, 3 and then prove it for all n.
n r’1 n’r’1 n’1
n
(x ’ j) (y ’ k) = (x + y ’ q).
r
r=0 q=0
j=0 k=0

Exercise K.242. [11.5, T, ‘ ] The following is a modi¬cation of Cauchy™s
proof of the binomial theorem. We shall work in R. We take w as a ¬xed
real number with |w| < 1.

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