# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$L=Lim ... 1answer 24 views ### Convergence of “sliced” power series Let \phi(t)=\sum_{k=1}^\infty a_k t^k, x=t^m \in \mathbb{C} for some fixed m\in \mathbb{N} be a convergent power series. I guess that a_0=0. For r=0,\ldots,m-1 and k=mq+r, why are the ... 1answer 60 views ### \prod_{k=1}^n (1+ \frac{z}{k}) converges to 0 when \Re (z)<0 We have a complex number z such that \Re (z)<0 and the sequence z_n = \prod_{k=1}^n (1+ \frac{z}{k}). Prove that \lim_{n \rightarrow \infty} z_n = 0. How to do it? I guess it will be ... 1answer 45 views ### g_n(z)=z^n uniformly on D={z: |z|<1}? I am looking at this example: g_n(z)=z^n and domain D={z: |z|<1} I see that every g_n will converge to 0 for n \rightarrow \infty. Thus it converges. Now, how can I show that it is or ... 1answer 33 views ### Application of Stirling's theorem for the given series I want to prove whether x=-4/27 is convergent or not for the series$$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$I applied alternating series test. But, while using this, I need to apply Stirling's ... 1answer 24 views ### The convergence interval of the series I want to prove whether x=4/27 is convergent or not for the series$$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$I used Raabe's test. But I got limit is 1. So the test is not valid. Please help me ... 3answers 114 views ### Please more help me to find the convergence interval and the sum -by using residue theory- of the series. The sum is that$$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test.$$ ... 2answers 146 views ### Sum of analytic functions converges uniformly but not the product We know that if$\sum_{n=1}^\infty |f_n(z)|$converges uniformly on$S$to a bounded function and$f_n(z)\neq -1$on$S$then$\prod_{n=1}^\infty (1+f_n(z))$converges uniformly on$S$. I want a ... 2answers 157 views ### For what real$r$does$\sum_{n=2}^\infty\frac1{n(\ln n)^{1+ir}}$converge and for which$r$does the sum diverge? A standard convergence result is that$\sum_{n=2}^\infty\frac1{n(\ln n)^p}$converges for real$p > 1$and diverges for$p \le 1$. How about complex$p$with real part$1$? Specifically, for what ... 1answer 180 views ### Proving that$\sum (-1)^{n+1} n^{-z}$defines an analytic function in$Re z>0$I want to show that the series$\sum_{n=1}^\infty (-1)^{n+1} n^{-z}$converges to an analytic function for$\Re z>0$. For$\Re z>1$the terms are dominated by$n^{-x}$so that we have absolute ... 0answers 82 views ### Infinite products (involving complex numbers) I am learning the Gamma function, based on some lecture notes, and I wish to ask a couple of questions regarding infinite products. Let$z$be a complex number except$\{0, -1, \ldots \}$. (1) How ... 1answer 62 views ### Calculate the Radius of convergence of$\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$I need your help: Calculate the Radius of convergence of the following: $$\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$ Im new to this subject, so I'd appreciate it if you can add explanations to ... 1answer 48 views ### Finding the values of$z$s.t.$\sum_{n=0}^{\infty} \left( \frac{z}{1+z} \right)^n$is convergent I manipulated the series to$\sum_{n=0}^{\infty} \left( \frac{1}{1-(-1/z)} \right)^n$, which converges for$|-1/z|<1$by geometric series. Then solving for$z$, I obtained$z>(1/\bar{z})$. Is ... 4answers 2k views ### Radius of Convergence of power series of Complex Analysis I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form$a_nz^n$but don't have a clue what to do with these. Any help would be ... 3answers 115 views ### showing$1 + z + z^2 + \dots $uniformly converges to$\frac{1}{1-z}$for$|z| < 1$What test can I use to show that$1 + z + z^2 + \dots $uniformly converges to$\frac{1}{1-z}$for$|z| < 1$. I know$\displaystyle 1 + z + z^2 + \dots +z^n = \frac{1-z^{n+1}}{1-z}$and as$n \to ...
Show that the roots of $$f(z) = z^n+z^3+z+2 =0$$ converge to the circle $|z|=1$ as $n \to \infty$.
I have the following sequence of function: $$f_n(\lambda)=\bigg[\alpha-i\bigg(\lambda+\frac{1}{n}\bigg)\bigg]^{-1}-\bigg[\alpha-i\lambda\bigg]^{-1},\,\,\,\alpha\neq 0$$ and I have to study its ...