1
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1answer
39 views

convergence radius of taylor series of a complex function in different directions, the same?

Given the taylor expansion of a complex functionf(z) around $z_0$, is the convergence radius of this series the same in different directions, say in real axis ...
0
votes
0answers
31 views

series functions of complex variable $ z $ and alternating zeta function convergence

Let $f_n(z)$ and $p_n(z)$ two series functions of complex variable $z$ defined as the following: $f_n(z)$=$ \sum_{n}exp({(-1)}^{n-1}{n^{-z}})) $ $p_n(z)$=$exp(\sum_{n}({(-1)}^{n-1}{n^{-z}})$ ...
0
votes
1answer
24 views

Holomorphic function and series converges in the unit disk $ (|z_{k}| < 1) $

$ f $ is holomorphic in the unit disc , bounded and not identically zero and $z_{1},z_{2},\ldots,z_{n},\ldots $ are its zeros$ (|z_{k}| < 1) $ , $a$ is a real number My question is :for which ...
3
votes
0answers
69 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
1
vote
1answer
76 views

Quick question on convergence

Shouldn't the radius of convergence be defined as: $$\frac{1}{R} = \lim_{n\rightarrow \infty} \left(a_n \right)^{\frac{1}{n}}$$ Not sure what they are doing above..
2
votes
2answers
68 views

When is $\sum_{n \ge 0} g_n(z)$ analytic?

Let $D$ be an open subset of $\mathbb{C}$ where $g_n(z)$, $n \in\mathbb{N}$ are analytic. Then $$f(z)=\sum_{n \ge 0}g_n(z)$$ is analytic on $D$ iff $\sum_{n\ge 0}g_n(z)$ is locally uniformly ...
0
votes
1answer
61 views

Analysis of singularities and taylor representation of $f(z)=\frac{z^2-1}{\sin \pi z}$

Let $$f(z)=\frac{z^2-1}{\sin \pi z}$$ A) Find all singulartities of $f$ in $\mathbb{C}$ and classify each as a pole (specifying the order), essential, removable, or other. B) Explain why $f(z)$ has ...
0
votes
1answer
70 views

Convergence of $\frac{1}{2^k} \frac{1}{z-w_k}$

Suppose $w_1,w_2,w_3,...$ are points on the unit circle. Consider the infinite series $$\sum_{k=1}^{\infty} \frac{1}{2^k} \frac{1}{z-w_k}$$ Let $D=\{z \in \mathbb{C}: |z|<1 \}$ A) Show that series ...
0
votes
2answers
76 views

What's the behavior of $\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$ outside its radius of convergence?

I want to check the behavior of $$\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$$ outside its radius of convergence. I've tried to use the ratio test as follows: ...
1
vote
1answer
116 views

How to show $\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$

For $z\in\mathbb{R}$ it's very easy to show that it holds $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$$ But how do we show the same thing for $z\in\mathbb{C}$
1
vote
2answers
236 views

Convergence of infinite product $\prod_{n=2}^\infty (1- \frac 1n) $

I am revising Complex Analysis and I am a bit confused. I have a couple of results from lectures which say that $\prod_{n=1}^\infty (1+a_n)$ converges if and only if the sum $\sum_{n=1}^\infty ...
1
vote
1answer
22 views

$\left(\frac{a_n}{n^k}\right)_n$ is bounded implies $\sum_{n=0}^\infty a_nz^n$ has a radius of convergence $\ge 1$

Let $$\left(\frac{a_n}{n^k}\right)_n\subset\mathbb{C}\;\;\;\;\;(k\in\mathbb{N})$$ be a boundet sequence. I want to show that the power series $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
0
votes
1answer
43 views

Radius of Convergence in Complex Analysis. [closed]

Following Questions are asked in previous years university exams. I'm preparing for the same exam to be held in next month. Please help me to solve these problems. I have no idea how to solve these ...
2
votes
1answer
71 views

Prove that $\sum_{n=0}^{\infty}e^{in\theta}$ is bounded

For my homework class, we need to prove that a certain series converges, for which it is useful to use that this series is bounded ($\theta \in (0,2\pi)$): $$\sum_{n=0}^{\infty}e^{in\theta}$$ I ...
0
votes
3answers
49 views

How to calculate $\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}$ with $z\in\mathbb{C}$ and $|z|\ne 1$?

As stated in the title: How does one calculate $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}$$ with $z\in\mathbb{C}$ and $|z|\ne 1$?
2
votes
0answers
40 views

When to Interchange Limit & Integral

I got stuck in the proof of Cauchy's Integral Formula for higher derivatives: Under what Conditions over a function $f$, we can infer that : $\displaystyle\lim_{h\rightarrow ...
1
vote
1answer
44 views

Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
1
vote
1answer
36 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
0
votes
1answer
36 views

Finding the radius of convergence of a series with only even powers.

I have the series $\displaystyle\sum_{n=0}^{\infty} \dfrac{z^{2n}}{(2+i)^n}$, and I need to find its radius of convergence. I cannot think how to arrange this so I can find $ R= \lim ...
0
votes
1answer
25 views

Problems with proving that a sequence converges to some certain limit

So here is my problem, I would like the prove the following, For any $m=0,1,2...$ and for $\alpha\in \mathbb C$ with $\Re(\alpha)<0$ it holds that, $$\lim_{t\rightarrow\infty}t^me^{-\alpha ...
0
votes
2answers
41 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
-2
votes
3answers
52 views

Showing that the sequence $z^n$ is normally but not uniformly convergent [duplicate]

I was able to show that $z^n$ is normally convergent on the unit disk centred at the origin, but I am not sure how to show that it is not uniformly convergent on the unit disk centred at the origin. ...
0
votes
1answer
39 views

Showing that $\sum\limits_{n=1}^{\infty}f_n-f_{n+1}$ converges normally if $f_n$ converges normally

A function sequence $\{f_n\}$ converges normally in an open set $U \subseteq \mathbb{C}$ to the limit function $f$. Show that the series $\sum\limits_{n=1}^{\infty}(f_n-f_{n+1})$ converges normally ...
0
votes
1answer
36 views

Property of a normally convergence squence of functions

Let the terms of a function sequence $\{f_n\}$ be continuous in an open set $U \subseteq \mathbb{C}$. If $\{f_n\}$ converges normally in $U$ to the limit function $f$ and if $z_n \to z_0$ in $U$, ...
1
vote
1answer
74 views

sequence uniformly convergent on the boundary of a bounded set in $\mathbb{C}$

Let $(f_n)$ be a sequence of functions which are analytic on a bounded region $A\subset\mathbb{C}$ and continuous on the closure $Cl(A)$. Suppose that the sequence is uniformly convergent on the ...
1
vote
0answers
181 views

What does the convergence of a Dirichlet series tells us about the convergence of a power series?

If $D(s)=\displaystyle \sum_{k\geqslant 1} f(k)\, k^{s}$ converges for $\Re(s)\lt a$, what is the radius of convergence of $\displaystyle \sum_{k\geqslant 1}f(k)\, x^k$ $=T(x)$? Conversely, what ...
1
vote
1answer
23 views

Finding the Power Series of a Complex fuction.

Find a power series expression $\sum_{n=0}^\infty A_n z^n $ for $ \frac{1}{z^2-\sqrt2 z +2} $ I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ...
0
votes
1answer
54 views

absolute convergence of infinite product

We know if $\sum_{n=1}^\infty|z_n|$ converges then $\sum_{n=1}^\infty z_n$ converges absolutely. (kind of trivial) I wonder whether it holds for infinite products, that is, if ...
0
votes
0answers
28 views

convergence of two complex series

I want to know whether the following statement is true or not. $\sum_{n=1}^\infty \log(1+z_n)$ converges iff $\sum_{n=1}^\infty z_n$ converges. I know $\sum_{n=1}^\infty \log(1+z_n)$ converges ...
1
vote
1answer
51 views

Convergence question and degree of polynomial

I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: $1)$ Suppose the power series ...
1
vote
0answers
60 views

Normal convergence versus uniform convergence

I am wondering about the nature of uniform and normal convergence. I know that uniform convergence is a weaker condition than normal convergence and that normal convergence even implies uniform ...
0
votes
4answers
167 views

Ratio test: $n^\sqrt{n}$

I need to determine the radius of convergence of: $$\sum_{n=1}^\infty z^n n^\sqrt{n}$$ I have, by use of the ratio test, written: (Because I know it tends to 1) $$\lim_{n\to\infty} ...
3
votes
2answers
58 views

How to calculate radius of convergence of the following series?

How can I calculate radius of convergence of the following series? $$\Large \sum\limits_{n=0}^\infty \frac{5^{n+1}}{\sqrt[n]{(2n)!}}z^{n} $$ I tried using D'alembert convergence test but cannot ...
3
votes
3answers
111 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
0
votes
0answers
44 views

Behaviour of $\sum_{n=1}^{\infty}\frac{z^{n}}{n}$ for $|z| = 1$ [duplicate]

How to show that the power series $$\sum_{n=1}^{\infty}\frac{z^{n}}{n}$$ ( which has radius of convergence $1$ ) converges in all points of $\partial D(0,1)$ except $z = 1$ ?
3
votes
1answer
66 views

Comparison between infinite products and series

I need examples of the following facts: 1) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges $\nRightarrow \prod_{j=0}^{+\infty}(1+|a_{j}|)$ converges 2) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges ...
0
votes
1answer
21 views

Exponent of convergence of $\{m+in \ | \ m, \ n \in \mathbb{Z}, (m,n) \neq (0,0) \}$

The exponent of convergence $\rho$ of a sequence $\{z_{n}\} \subset \mathbb{C}$ is defined as $$\rho = \inf \{\lambda \geq 0 \ | \sum_{n}\frac{1}{|z_{n}|^{\lambda}} < +\infty\}$$ My doubt is : ...
1
vote
0answers
41 views

Range of values for which the integrals converge

I have two integrals (i) $\int_\gamma e^{z^2}dz$ where $\{\gamma: z=se^{i\alpha}: -\infty<s<\infty\}$ (ii) $\int_0^{\infty}\frac{x^\beta dx}{1+x}$ I know that the first integral converges for ...
0
votes
1answer
56 views

Series $\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}} \ $, $ z \in \mathbb{C}$

I'm studying the series $$\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}}$$ If $z = x+iy$, what is the behaviour of the series for $-1<x<0 \ $?
1
vote
2answers
76 views

Find the radius of convergence for $\sum^{\infty}_{n=0} a_nz^{n!}$

Find the radius of convergence for $\sum^{\infty}_{n=0} a_nz^{n!}$ where $a_0=2i$ and $a^2_n=a_{n+1}$ This is not a power series, but if I define $b_k=a_n$ if $k=n!$ for $n \geq 2$ (since $b_0 = 0$ ...
2
votes
2answers
77 views

Infinite product $(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})$

I have to show that if $|z| < 1$, $z \in \mathbb{C}$, $$(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})= \frac{1}{1-z}$$ I want to understand how to do this kind of exercises, any hint ?
1
vote
2answers
82 views

Infinite product convergence

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges. I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ ...
1
vote
2answers
70 views

Convergence radius: $R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid}$ (incl. $\infty$) when $R = 0$ and Ratio test

I have read the following proof of a theorem in a textbook of mine, and I've been wondering why the proof holds when $$R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid} = 0$$ ...
1
vote
2answers
98 views

Uniform convergence of a sequence of holomorphic polynomials

This is a two-part question : Let $E = \{|z|=1 \text{ and } \Im{z} \geq 0\}$. Does there exist a sequence of (holomorphic) polynomials in $z$ which converge uniformly on $E$ to $f(z) = \bar{z}$? ...
3
votes
2answers
87 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
1
vote
1answer
98 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
4
votes
1answer
53 views

$\sum_{n=1}^\infty\frac{z^2}{1+n^2z^2}$ converges to analytic function

For which $z$ does $\sum_{n=1}^\infty\dfrac{z^2}{1+n^2z^2}$ converge to an analytic function? What are its poles? I think the poles should be $\pm\dfrac{i}{n}$, since those are the values at ...
2
votes
1answer
43 views

Uniform convergence of product in ball $|z|<R$

Find a polynomial $P(z)$ such that $$\prod_{n=1}^\infty \left(1+\dfrac{z}{\sqrt{n}}\right)e^{-P(z)}$$ converges uniformly in every ball $|z|<R$. Maybe we should take log to get that the sum ...
3
votes
1answer
38 views

Limit of product series for convergent and increasing sequences

Suppose $a_1,a_2,\ldots\in\mathbb{C}$ and $b_1,b_2,\ldots\in\mathbb{R}$. Suppose also that $\sum a_n$ converges, that $b_n\leq b_{n+1}$ for all $n\geq 1$, and that ...
2
votes
1answer
89 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...