Section 5.2

Series and Sequences of Functions

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Problem 5.2.1

Show that

(1+z+z2+⋯+z9)(1+z10+z20+⋯+z90)(1+z100+z200+⋯+z900)⋯=11−z (1+z+z^{2}+\cdots +z^{9})(1+z^{10}+z^{20}+\cdots +z^{90})(1+z^{100}+z^{200}+\cdots +z^{900})\cdots = \frac{1}{1-z}

for ∣z∣<1\left|z\right|<1.

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Problem 5.2.2

Suppose the coefficients of the power series

∑n=0∞anzn \sum _{n=0}^{\infty } a_{n} z^{n}

are given by the recurrence relation

a0=1,a1=−1,3an+4an−1−an−2=0,n=2,3,…. a_{0}=1, \quad a_{1}=-1, \quad 3a_{n}+4a_{n-1}-a_{n-2}=0, \quad n=2,3,\ldots .

Find the radius of convergence of the series and the function to which it converges in its disc of convergence.

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Problem 5.2.3

Show that the differential equation

f′′(z)=zf(z),f(0)=1,f′(0)=1 f''(z) = zf(z), \qquad f(0)=1, \qquad f'(0)=1

has an unique entire solution in the complex plane.

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Problem 5.2.4

Describe the region in the complex plane where the infinite series

∑n=1∞1n2exp⁡(nzz−2) \sum _{n=1}^{\infty } \frac{1}{n^{2}} \exp \left(\frac{nz}{z-2}\right)

converges. Draw a sketch of the region.

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Problem 5.2.5

Let ff be an analytic function such that

f(z)=1+2z+3z2+⋯for ∣z∣<1. f(z) = 1+2z+3z^{2}+\cdots \quad \text{for } \left|z\right|<1.

Define a sequence of real numbers a0,a1,a2,…a_{0}, a_{1}, a_{2}, \ldots by

f(z)=∑n=0∞an(z+2)n. f(z) = \sum _{n=0}^{\infty } a_{n}(z+2)^{n}.

What is the radius of convergence of the series

∑n=0∞anzn? \sum _{n=0}^{\infty } a_{n} z^{n}?
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Problem 5.2.6

Let the sequence a0,a1,…a_{0}, a_{1}, \ldots be defined by the equation

1−x2+x4−x6+⋯=∑n=0∞an(x−3)n(0<x<1). 1-x^{2}+x^{4}-x^{6}+\cdots = \sum _{n=0}^{\infty } a_{n}(x-3)^{n} \quad (0<x<1).

Find

lim sup⁡n→∞(∣an∣1n). \limsup _{n\to \infty } \left(\left|a_{n}\right|^{\frac{1}{n}}\right).
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Problem 5.2.7

Suppose the power series

∑n=0∞anzn \sum _{n=0}^{\infty } a_{n} z^{n}

converges for ∣z∣<R\left|z\right|<R where zz and the ana_{n} are complex numbers. If bn∈Cb_{n} \in \mathbb {C} are such that ∣bn∣<n2∣an∣\left|b_{n}\right| < n^{2}\left|a_{n}\right| for all nn, prove that

∑n=0∞bnzn \sum _{n=0}^{\infty } b_{n} z^{n}

converges for ∣z∣<R\left|z\right|<R.

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Problem 5.2.8

Let b1,b2,…b_{1}, b_{2}, \ldots be a sequence of real numbers such that bk≥bk+1b_{k} \geq b_{k+1} for all kk and lim⁡k→∞bk=0\lim_{k\to \infty } b_{k}=0. Prove that the power series ∑k=1∞bkzk\sum_{k=1}^{\infty } b_{k} z^{k} converges for all complex numbers zz such that ∣z∣≤1\left|z\right| \leq 1 and z≠1z \neq 1.

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Problem 5.2.9

For which z∈Cz \in \mathbb {C} does

∑n=0∞(znn!+n2zn) \sum _{n=0}^{\infty } \left(\frac{z^{n}}{n!} + \frac{n^{2}}{z^{n}}\right)

converge?

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Problem 5.2.10

Show that

∑n=0∞z(1+z2)n \sum _{n=0}^{\infty } \frac{z}{(1+z^{2})^{n}}

converges for all complex numbers zz exterior to the lemniscate

∣1+z2∣=1. \left|1+z^{2}\right| = 1.
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Problem 5.2.11

Determine the complex numbers zz for which the power series

∑n=1∞znnlog⁡n \sum _{n=1}^{\infty } \frac{z^{n}}{n^{\log n}}

and its term by term derivatives of all orders converge absolutely.

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Problem 5.2.12

Suppose

f(z)=∑n=0∞anzn f(z) = \sum _{n=0}^{\infty } a_{n} z^{n}

has radius of convergence R>0R>0. Show that

h(z)=∑n=0∞anznn! h(z) = \sum _{n=0}^{\infty } \frac{a_{n} z^{n}}{n!}

is entire and that for 0<r<R0<r<R, there is a constant MM such that

∣h(z)∣≤Me∣z∣/r. \left|h(z)\right| \leq Me^{\left|z\right|/r}.
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Problem 5.2.13

Let the power series ∑n=0∞cnzn\sum_{n=0}^{\infty } c_{n} z^{n}, with positive radius of convergence RR, represent the function ff in the disk ∣z∣<R\left|z\right|<R. For k=0,1,…k=0,1,\ldots let sks_{k} be the kkth partial sum of the series sk(z)=∑n=0kcnzns_{k}(z) = \sum_{n=0}^{k} c_{n} z^{n}. Prove that

∑k=0∞∣f(z)−sk(z)∣<∞ \sum _{k=0}^{\infty } \left|f(z)-s_{k}(z)\right| < \infty

for each zz in the disk ∣z∣<R\left|z\right|<R.

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Problem 5.2.14

Let R>1R>1 and let ff be analytic on ∣z∣<R\left|z\right|<R except at z=1z=1, where ff has a simple pole. If

f(z)=∑n=0∞anzn(∣z∣<1) f(z) = \sum _{n=0}^{\infty } a_{n} z^{n} \qquad (\left|z\right|<1)

is the Maclaurin series for ff, show that lim⁡n→∞an\lim_{n\to \infty } a_{n} exists.

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Problem 5.2.15

Find the radius of convergence RR of the Taylor series about z=1z=1 of the function f(z)=1/(1+z2+z4+z6+z8+z10)f(z) = 1/(1+z^{2}+z^{4}+z^{6}+z^{8}+z^{10}). Express the answer in terms of real numbers and square roots only.

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Problem 5.2.16

Prove that the uniform limit of a sequence of complex analytic functions is complex analytic. Is the analogous theorem true for real analytic functions?

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Problem 5.2.17

Let gn(z)g_{n}(z) be an entire function having only real zeros, n=1,2,…n=1,2,\ldots. Suppose

lim⁡n→∞gn(z)=g(z) \lim _{n\to \infty } g_{n}(z) = g(z)

uniformly on compact sets in C\mathbb {C}, with gg not identically zero. Prove that g(z)g(z) has only real zeros.

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Problem 5.2.18

Let f,g1,g2,…f, g_{1}, g_{2}, \ldots be entire functions. Assume that

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(1)

∣gn(k)(0)∣≤∣f(k)(0)∣\left|g_{n}^{(k)}(0)\right| \leq \left|f^{(k)}(0)\right| for all nn and kk;

(2)

lim⁡n→∞gn(k)(0)\lim_{n\to \infty } g_{n}^{(k)}(0) exists for all kk.

Prove that the sequence {gn}\left\{ g_{n}\right\} converges uniformly on compact sets and that its limit is an entire function.

Problem 5.2.19

Find a Laurent series that converges in the annulus 1<∣z∣<21<\left|z\right|<2 to a branch of the function log⁡(z(2−z)1−z)\log \left(\dfrac {z(2-z)}{1-z}\right).

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