1
vote
2answers
68 views

If a series converges then the power series converges for all z

How can I prove that if $\sum \limits_{n=1}^{\infty} c_n$ , $c_n\in \mathbb{C}$, converges then $\sum \limits_{n=1}^{\infty} c_n \frac{z^n}{1-z^n}$ converges for all z in $\mathbb{C}$ with ...
2
votes
2answers
97 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
1
vote
3answers
61 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
0
votes
2answers
62 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
1
vote
0answers
43 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
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!
0
votes
1answer
48 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
1
vote
2answers
46 views

Find radius of convergence of power series

Since we know that given $\sum_{n=0}^{\infty }C_nz^n$, if $\lim_{n\rightarrow \infty }|C_n|^{1/n}$ exists then $R^{-1}=\lim_{n\rightarrow \infty }|C_n|^{1/n}$ where $R$ is the radius of convergence. ...
2
votes
1answer
58 views

Complex Exponential/Trigonometric Functions

I'm having trouble on proving the following state of a Lemma using the power series of $\exp z$ centered at $0$: For all $z \in C$: $\exp(z + 2\pi i) = \exp(z)$ and $\exp(z) \neq 0$ All help ...
3
votes
3answers
113 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
2answers
27 views

Trouble on proving a lemma - Complex Power Series

I'm having some difficulties on proving the following lemma: "If $f_n$ is a sequence of functions which converges uniformly to $0$ on a set $G$ and $z_n$ is any sequence in $G$ then the sequence ...
1
vote
2answers
57 views

Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
1
vote
0answers
40 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 ...
2
votes
1answer
54 views

Want to check analyticity of a series on a open disk.

How do we check the analyticity of a any power series? For example: How will we show that $$f(z):= \sum_{n=1}^\infty z^{n!}= z^1+z^2+z^6+z^{24}......+z^{n!}......$$ is anaytic on disk {$z : ...
8
votes
6answers
493 views

Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + … + r^{n-1} = 0$.

Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + ... + r^{n-1} = 0$. I think I can represent all the roots of 1 as follows: $r = 1^{\frac{1}{n}} ( \frac{\cos{2\pi k}}{n} + ...
2
votes
1answer
207 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
4
votes
4answers
246 views

Convergence of $\sum_{n=0}^{\infty}\dfrac{z^n}{1+z^{2n}}$

For what complex values of $z$ is $$\sum_{n=0}^{\infty}\dfrac{z^n}{1+z^{2n}}$$ convergent? I would like to write the sum as a power series, because with a power series we can determine the radius ...
1
vote
1answer
230 views

Radius of convergence of power series (complex)

I don't know if my reasoning is right on this exercise: If the power series $\sum a_n z^n$ has radius of convergence $R$, which is the radius of convergence of the series $\sum a_n^2 z^n$ and $\sum ...
8
votes
5answers
263 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
0
votes
1answer
80 views

Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.

I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
2
votes
1answer
149 views

Prove the following equation of complex power series.

Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have $$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$ $$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$ My guess ...
0
votes
1answer
179 views

When - during the study and development of- - and how were complex numbers introduced in the study of [real-valued] power series (expansions)?

Follow-up to: Mathematical reason for the validity of the equation: $S=1+x^2S$ and General question on relation between infinite series and complex numbers (This question seems broad at this stage, ...