All Questions
16
questions
1
vote
1
answer
123
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Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
0
votes
1
answer
101
views
Strange summation? $\sum _{k=-n}^{n+1} \frac{(-1)^k}{x-k}$
I'm mainly concerned with the bounds of summation here. I've never personally seen such a summation before, but I came across this summation in "Special Functions" by Andrews Askey Roy on ...
0
votes
2
answers
52
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What is the Radius of Convergene of $f(z) = \sum_{n=0}^\infty \frac1{4^n}z^{2n+1}$
Here's what I have:
$f(z) = z + \frac14z^3+\frac1{4^2}z^5+...$
So, my coefficients are either $0$ or $\frac1{4^n}$ with $\frac1{4^n}$ being the supremum.
So, $\limsup \limits_{n \to \infty} |c_n|^{\...
- 70
1
vote
1
answer
116
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How do I obtain the Jacobi theta function?
If I enter the series $$\sum_{n=-\infty}^\infty e^{-(n+x)^2}$$ into WolframAlpha it expresses it by the Jacobi theta function $$\sqrt{\pi} \vartheta_3(\pi x,e^{-\pi^2})=\sqrt{\pi} \big(1+2\sum_{n=1}^\...
- 139
2
votes
1
answer
167
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On a alternate series representation of Riemann xi function
In a recent paper Dan Romik proved the following alternating infinite series representation for Riemann xi function:
I may be wondering if we can transform this infinite alternating sum into Abel ...
- 916
1
vote
0
answers
102
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Searching for closed-form solutions to integral of dilogarithm
while trying to evaluate an infinite sum, I came across this integral:
$$ \displaystyle \mathcal{I}=\int_{0}^{\infty}\frac{\sin(x)}{x}\DeclareMathOperator{\dilog}{Li_{2}}\dilog\left(-e^{-x}\right)\...
0
votes
4
answers
94
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zero's for odd and $\pm 1$ for even. Or $\pm 1$ for odd and zero's for even
What is the formula of the real function $f$ that satisfies
\begin{equation}
\sum^{n}_{k=0}{f}=1+0+(-1)+0+1+0+(-1)+0+\cdots
\end{equation}
or
\begin{equation}
\sum^{n}_{k=0}{f}=0+1+0+(-1)+0+1+0+(-1)...
- 61
1
vote
1
answer
177
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Pattern in Squared Numbers and their Digit Sum
So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
0
votes
1
answer
77
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Chapter V: Titchmarsh's book "The theory of the Riemann Zeta function"
Through chapter V of Titchmarsh's book "The theory of the Riemann Zeta function" it is used a "counting technique" that I am not understanding. In particular, Theorem 5.12, p 106, uses something like:...
- 355
3
votes
0
answers
85
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Replacing $n!$ with Stirling's approximation in $e^x = \sum_n \frac{x^n}{n!}$
I was wondering if there is a closed-form expression for
$$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$
although I expect there is none because Mathematica cannot compute it. However, from Stirling'...
- 778
1
vote
3
answers
123
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Suppose $(a_i)_{i=1}^{\infty}$ is square summable. Is $(\frac{a_i}{i})_{i=1}^\infty$ absolutely summable?
Suppose $(a_i)_{i=1}^{\infty}$ is square summable. Is $(\frac{a_i}{i})_{i=1}^\infty$ absolutely summable?
Asked another way, say we have that $\sum a_i^2 \lt \infty$. Does this imply that $\sum|\frac{...
- 29
1
vote
1
answer
57
views
Order of an infinite sum
How to prove that, for $0<c<1$,
$$
\sum\limits_{j=1}^{\infty} c^{\, j } \cdot j^{\, -(\frac{d}{2} +1 )}
$$
is, for some positive constant $K$, of order $K + O( \, ( 1-c )^{\frac{d-2}{2}})$ when ...
- 200
2
votes
1
answer
235
views
How to solve this equation with implicit sum
I want to know how the authors of this arxiv paper (p. 10) solved the equation
\begin{align}
g\left(\lambda\right)
={}&
\frac{1}{2\pi}\sum_{\omega\in\left[0,2\pi\right]:f\left(\omega\right)=\...
- 167
3
votes
1
answer
124
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Please calculate $\sum _{ k=0 }^\infty\left[ \tan^{ -1 }\left( \frac { 1 }{ k^{ 2 }+k+1 } \right) -\ldots \right] $
Not many math problems stump me, but this summation has me stumped. Can someone provide a solution to this summation:
$$\sum _{ k=0 }^{ \infty }{ \left[ \tan ^{ -1 }{ \left( \frac { 1 }{ k^{ 2 }+k+1 }...
- 91
3
votes
1
answer
325
views
$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$
Hi I am trying to calculate the sum given by
$$
\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} \vartheta_3\big(-\frac{\pi\beta}{2\alpha},e^{-\pi^2/(...
- 9,966