# Questions tagged [exponential-sum]

For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.

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### How to show $\displaystyle\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$

This result seems trivial, how would I show $$\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$$ Where $0 \le k,l \le n-1$ And $k \neq l \neq 0$. I tried to use the exponential definition ...
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### Limit of summation of $\sum_{i=0}^{\infty} e^{-ik}$ series

I was wondering whether anyone was able to give some insight to determining the limit of the following summation $$\sum_{i=0}^{\infty} e^{-ik}$$ If anyone has any idea or is able to point me in the ...
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### Modified exponential summation [duplicate]

How do we prove that $$1+\frac{2^3}{2!}+\frac{3^3}{3!}+\frac{4^3}{4!}+\cdots=5e$$ I'll post the answer to this question as a knowledge share.
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### Is there a closed form for $\sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y}$?

I would like to find a closed formula for this equation: $$\sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y}$$ Both the denominator and also the exponent is changing in each step. How is it possible ...
1 vote
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### Bounds on $\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$?

Anyone familiar with this exotic infinite sum? I'm assuming $n$ and $x$ are positive integers as well as $n-x>1$ (for convergence). $$\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$$ One can ...
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### Complex representation of a real wave

I have the following formula for a wave: $u(x)=a \cos(kx-\omega t + \phi)$. I am trying to recover the complex representation of it $u(x)=\Re(a e^{i\phi}e^{i(kx-wt)}$. I started by using Euler's ...
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### Infinite sum over $e^{i \frac{\pi (n-x)^2}{a}}$

I would like to evaluate the infinite sum $$\sum_n \exp \left( \frac{i\pi}{a} (n-x)^2\right)$$ where $x,a\in \mathbb R$. For certain values of $a$ it becomes a Delta comb of various frequencies, ...
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### $|\sum\limits_{k\in \mathbb{Z}}P(X=k)e^{ikt}|=1$ implies $X$ is almost surely constant

Question : Let's suppose $P$ is a probability, $X$ a random variable defined on $\mathbb{Z}$. Let's suppose we have $|\sum\limits_{k\in \mathbb{Z}}P(X=k)e^{ikt}|=1$ for all $t\in \mathbb{R}$. Prove ...
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I'd like to have a closed formula for $\sum\limits_{i=0}^n e^{\sqrt{i}x}=1+e^{\sqrt{1}x}+...+e^{\sqrt{n}x}$, something similar to the formula $\sum\limits_{i=0}^ne^{ix}=\frac{e^{(n+1)x}-1}{e^x-1}$. ...
let's define $f:\mathbb{R} \to \mathbb{C}$ a $2\pi$ periodic continuous function. Let's say $s_{n}(x)=\sum_{k=-n}^{n} f\hat (k) e^{ikx}$. Let's say $t_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n} s_{k}(x)$. ...