# Questions tagged [exponential-sum]

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

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### Proof that $\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$

I am looking for an analytical proof that : $$\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$$ Both CAS Wolfram alpha and sympy agree on the result : ...
1 vote
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### Analytical evaluation of infinite series

I am trying to calculate the infinite series $$\sum _{n=-\infty }^{\infty } \frac{(-1)^{n+1} e^{-(n-1)^2\pi}}{1-e^{ (2 n-1)\pi}}\simeq -0.0903244354808$$ Are there any any analytical methods to ...
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### Exponential function to the power of a fraction

I have an equation such that $$1=e^{\frac{ylog(3+x)}{3+x}}-3 -x$$ Where e is the exponential function. How do I solve for x? According to the laws of exponents, $a^{m/n}=\sqrt[n]{a^m}$, Applying this ...
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### How to efficiently compute $\sum_{i=1}^n (x_i) ^ a$ for many different values of a and large n?

I need to compute $\sum_{i=1}^n (x_i) ^ a$ for many different values of a where n is large and x is fixed. I am in an environment where computing exponentials $(x_i) ^ a$ is effectively constant time. ...
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### Exponential sums reference

Looking for reference on exponential sums, in particular Jacobi, Gauss, Kloosterman and Ramanujan sums. The books mentioned in https://mathoverflow.net/questions/65429/exponential-sums-for-beginner ...
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### An infinite sum of products

I have to calculate this sum in closed form $$\sum_{n=1}^\infty \prod_{k=1}^n \frac{x^{k-1}}{1 - x^k}$$ where $x < 1$. Numerical evaluation shows that this converges. The product can be performed ...
1 vote
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### Bounding exponential sums

There is classical inequality that seems to often appear: $$\sum_{i=1}^N \exp(2i\pi n \theta) \leq \|\theta\|^{-1}$$ where $\|\theta\|$ denotes the distance to the closest integer. I do understand ...
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### Double EMA into Single EMA

The exponential moving average (EMA) operator is defined as: $$y_t(x, \lambda) = (1-\lambda) \sum_{i=0}^\infty \lambda^i x_{t-i}$$ where $1-\lambda$ is the normalization factor, and the operator is a ...
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### Product representation of some exponential functions?

I was looking at some old questions of mine and stumbled upon this quesiton, which I could not solve: Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The ...
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### Summation of Exponential

The equation below is a posteriori density under the assumption that the prior distribution for $\mu$ is normal. And I wonder why do the highlighted portions where $\mu$ and $x_{k}$ are swapped when ...
$$c^1 + c^2 + \cdots + c^n = c\frac{c^n - 1}{c-1}$$ Right? What about $c^\frac{1}{m} + c^\frac{2}{m}+ ... c^\frac{n}{m}$? Does it also have a closed formula? Thanks a lot!