All Questions
Tagged with sequences-and-series complex-analysis
2,026
questions
2
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0
answers
88
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Deriving the value of the series $\sum _{n=1}^{\infty }\left[\frac{2n}{e^{2\pi n}+1}+\frac{2n-1}{e^{\left(2n-1\right)\pi }-1}\right]$
I have been working on trying to show that
$\displaystyle \sum _{n=1}^{\infty }\left[\frac{2n}{e^{2\pi n}+1}+\frac{2n-1}{e^{\left(2n-1\right)\pi }-1}\right]=\frac{\varpi ^2}{4\pi ^2}-\frac{1}{8}$
...
- 65
1
vote
0
answers
67
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Theorem requirements not satisfied but answer is correct?
In Ron Gordon's answer here, he uses the residue theorem to compute the sum
$$\sum_{k=-\infty}^{\infty} \frac{(-1)^{k} (2 k-1)}{k^2-k+1}$$
The theorem for the case of infinite sums states that
Let $f ...
- 3,515
0
votes
1
answer
35
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Termwise Differentiation of Convergent Series Involving the Riemann Zeta Function
Let $\zeta(s)$ be the Riemann zeta function. We know that
$$\zeta(s) = \prod_p(1 - p^{-s})^{-1}, \ \operatorname{Re} s > 1$$
in the sense that the righthand side converges absolutely to $\zeta(s)$ ...
- 4,429
1
vote
0
answers
46
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Construct a sequence of complex numbers $z_n$ such that $z_1=1$ and $S:=\displaystyle\sum_{n=1}^\infty z_n^k = 0$ for every $k \in \mathbb{N}$. [duplicate]
Construct a sequence of complex numbers $z_n$ such that $z_1=1$ and $S:=\displaystyle\sum_{n=1}^\infty z_n^k = 0$ for every $k \in \mathbb{N}$.
I have some preliminary ideas. For now, fix $k=1$. Let $...
- 2,782
3
votes
1
answer
131
views
How to Prove Convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in \mathbb{C}$?
For $\frac{\sin(\pi z)}{\pi z} =\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$, prove convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in \mathbb{C}$.
...
- 6,620
0
votes
0
answers
31
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Absolutely vs Non-Absolutely Convergent Infinite Product
Consider the following standard infinite product:
$$
\prod_{n=1}^{+\infty} \left( 1 + \frac{(-1)^n}{2n-1}z \right)
$$
This product is not absolutely convergent because:
$$
\sum_{n=1}^{+\infty} \left|...
- 1,179
2
votes
0
answers
40
views
Cauchy product of absolutely convergent series
In Lang "Complex Analysis" (page 60) he gives a lengthy proof that if the power series $f= \sum a_n z^n, g = \sum b_n z^n$ both absolutely converge in some disc, then also their Cauchy ...
- 51
2
votes
0
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56
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Can I substitute $\cos^{(k)}(0)+i\sin^{(k)}(0)$ for $i^k$ in a series with index k?
I was trying to come up with a derivation of Euler's Formula inspired by a section in Thomas' Calculus, and I am using the following equation in this derivation
$$\sum_{k=0}^{\infty}\left(\frac{\cos^{(...
10
votes
2
answers
378
views
Associativity of infinite products
It is well-known that if $\sum_{n=1}^\infty a_n$ is an absolutely convergent complex series and $\mathbb N$ is partitioned as $J_1,J_2,\dots$, then the series $\sum_{j\in J_n}a_j$ for all $n$ and $\...
- 1,434
0
votes
0
answers
30
views
Estimation of the absolute value of an infinite sum by its coefficient
I got this problem from a complex funtion, but it now has little to do with the complex analysis.
$$\Theta(s)=\sum_{n=0}^{\infty}a_{2n}s^{2n}$$
I already have:
$$c_1^n\frac{(\ln n)^{2n}}{(2n)!}\leq |...
1
vote
0
answers
33
views
maximal continuation of $\Pi_2(x)$
Consider the functions for $k\in \Bbb N$
$$ \Pi_k(x) := \sum_{n \in \Bbb N} e^{\frac{\log^k n}{\log x}} $$
$\Pi_1(x)$ converges for real $1/e<x<1$.
$\Pi_1(x)$ is a Riemann zeta function i.e. $\...
- 754
0
votes
0
answers
32
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Classifying a singularity (how to find limit)
I have $f(z) = \frac{e^{-z}sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$
I already found that $z=2$ and $z=-2$ are poles of order $1$.
To classify $z=1$ I am struggling. I wrote $f(z) = \frac{1}{(z-1)^2} \sum_{n=0}^...
- 247
0
votes
1
answer
27
views
Finding largest value $R$ for Laurent Series
Find the largest value $R$ such that the laurent series of $\frac{2}{z^2-1} + \frac{3}{2z-i}$ about $z=1$ converges for $0 < |z-1| < R$
I observed that the isolated singularities are at $z=1,-1,\...
- 247
0
votes
1
answer
39
views
Question about convergence of a sequence of functions from $\Gamma$ [closed]
Let $\rho(z) = \big|\frac{z-i}{z+i}\big|$. If $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})$, then we have that $\left(\rho\left(\frac{ai+b}{ci+d}\right)\...
- 1,021
1
vote
2
answers
86
views
Manipulating a sum
I have the following sum:
$$ S = \sum_{n=1}^{+\infty} \frac{\cos(n\pi)}{(n+1)^2}$$
I'm asked to solve it using the following result:
Let $f(z)$ be a complex function with it's poles $z_{k} \not\in \...
- 47