# Tagged Questions

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### 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 ...
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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 ...
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### 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 ...
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 $$... 0answers 43 views ### Test the uniform convergence of the series in indicated region Test the uniform convergence of the series I tried to find M_n such that |\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n  by using Abel's Theorem This is the question : Test the ... 2answers 41 views ### Convergence of complex power series question I need some help to solve this problem and find the domain of convergence of the following power series:$$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$Thank you! 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 ... 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. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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!}......$$is anaytic on disk {z : ... 6answers 493 views ### Show that if r is an nth root of 1 and r\ne1, then 1 + r + r^2 + … + r^{n-1} = 0. Show that if r is an nth root of 1 and r\ne1, then 1 + r + r^2 + ... + r^{n-1} = 0. I think I can represent all the roots of 1 as follows: r = 1^{\frac{1}{n}} ( \frac{\cos{2\pi k}}{n} + ... 1answer 207 views ### Intuition regarding Taylor series for \frac{e^z}{1-3z}. The question asks me to find the Taylor series for$$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 ... 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}}$$convergent? I would like to write the sum as a power series, because with a power series we can determine the radius ... 1answer 230 views ### Radius of convergence of power series (complex) I don't know if my reasoning is right on this exercise: If the power series \sum a_n z^n has radius of convergence R, which is the radius of convergence of the series \sum a_n^2 z^n and \sum ... 5answers 263 views ### When are we (not) allowed to replace x by ix? It seems to be quite a common manipulation to replace x by ix. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ... 1answer 80 views ### Demonstrating the coefficients of the power series of \frac{1}{1-z-z^2} satisfies a recurrence relation. I have the power series$$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$and I'd like to show that the coefficients of this power series satisfy c_n=c_{n-1}+c_{n-2}. I thought the most obvious way ... 1answer 149 views ### Prove the following equation of complex power series. Show that for |z| \lt 1 with z \in \Bbb C, we have$$ \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 ...
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, ...