2
votes
0answers
30 views

When to Interchange Limit & Integral

I really 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
41 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
24 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
33 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
32 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
46 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
33 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
45 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
75 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
18 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
43 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
24 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
40 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
37 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
153 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
56 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
71 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 ...
1
vote
0answers
40 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
60 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
18 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
38 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
73 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
62 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
66 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
62 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
84 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
78 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
96 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
51 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
39 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
35 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
52 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 ...
3
votes
4answers
162 views

Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$

I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe ...
7
votes
1answer
122 views

$\Gamma(1/2-n+it)$ converges uniformly

Prove that $\Gamma(1/2-n+it)\rightarrow 0$ uniformly as $n\rightarrow\infty$ for $t\in\mathbb{R}$, where $n$ is a positive integer. I'm not sure which definition of $\Gamma$ would be easiest to ...
2
votes
1answer
38 views

Product with exponential converges absolutely and uniformly

Prove that the product $$\prod_{n=1}^\infty \left(1+\frac{z}{n}\right)e^{-z/n}$$ converges absolutely and uniformly on every compact set. What can I transform this product into? It's ...
0
votes
1answer
35 views

Complex power series (or not quite so?)

I'm stuck with this problem. Any hints are appreciated. It just says $$ \mbox{"For what values of}\ z\ \mbox{is}\quad \sum_{n = 0}^{\infty}\left(z \over 1+z\right)^{n}\quad \mbox{convergent ?} $$ ...
7
votes
1answer
241 views

Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda ...
1
vote
0answers
39 views

Where on the border of convergence circle series converges and where diverges?

I have power series of $ \sum\limits_{k=2}^{\infty} (\ln k)^{\alpha} z^k$. Alpha is a parameter. I've found the radius of convergence. R = 1. If $alpha \geq 0$ then series diverges for z from boundary ...
1
vote
0answers
26 views

When is meromorphic continuation possible?

Suppose I have an expression of the form $$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region ...
4
votes
2answers
69 views

A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof

I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a ...
0
votes
1answer
35 views

finding $\sum_{n=0}^\infty(\frac{(n+1)}{n})^{n^2}(z-2)^2 $ radius of convergence

find this power serie radius of convergence and the area where it converges. $\sum_{n=0}^\infty(\frac{(n+1)}{n})^{n^2}(z-2)^2 $ my attempt: a) $L=lim sup|an|^\frac{1}{n} \quad $$L=Lim ...
1
vote
1answer
26 views

Convergence of “sliced” power series

Let $\phi(t)=\sum_{k=1}^\infty a_k t^k$, $x=t^m \in \mathbb{C}$ for some fixed $m\in \mathbb{N}$ be a convergent power series. I guess that $a_0=0$. For $r=0,\ldots,m-1$ and $k=mq+r$, why are the ...
2
votes
1answer
66 views

$\prod_{k=1}^n (1+ \frac{z}{k})$ converges to $0$ when $\Re (z)<0$

We have a complex number $z$ such that $\Re (z)<0$ and the sequence $ z_n = \prod_{k=1}^n (1+ \frac{z}{k})$. Prove that $\lim_{n \rightarrow \infty} z_n = 0$. How to do it? I guess it will be ...
1
vote
1answer
45 views

$g_n(z)=z^n$ uniformly on $D={z: |z|<1}$?

I am looking at this example: $g_n(z)=z^n$ and domain $D={z: |z|<1}$ I see that every $g_n$ will converge to 0 for $n \rightarrow \infty$. Thus it converges. Now, how can I show that it is or ...
1
vote
1answer
33 views

Application of Stirling's theorem for the given series

I want to prove whether $x=-4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$ I applied alternating series test. But, while using this, I need to apply Stirling's ...
0
votes
1answer
25 views

The convergence interval of the series

I want to prove whether $x=4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$ I used Raabe's test. But I got limit is 1. So the test is not valid. Please help me ...
3
votes
3answers
115 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...