# Questions tagged [exponential-sum]

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

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### Can anyone help me calculate this sum

$$\sum_1^{n}a^{p^{n-1}}$$ I tried many things but couldn't find it. I think I'm missing something and that it might be related to the multinomial theorem.
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### Sum of powers of the sum of all the square numbers in a factorial

Here's the exact question: The sum of all positive integers $p$ such that $13! \div p$ is a perfect square, can be written as $2^a . 3^b . 5^c . 7^d . 11^e . 13^f$ where $a, b, c, d, e$ and $f$ are ...
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### Deriving an upper bound for an exponential function

I am studying risk theory this semester and we are currently covering the concept of adjustment coefficients, $R$. Essentially, the result we derived in class was that, for a random claim of size $x$, ...
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### How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$.

How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$? Here $\|\alpha x\|$ is the distance from $\alpha x$ to the nearest integer. My attempt was to rewrite ...
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### Proving that periodic vector-valued function satisfies $f(\mathbf x)= f(\mathbf x')$ iff $\mathbf x' = \mathbf x + k + \mathbf am$

In order to create a layer for a neural network that uniquely encodes relative relationships between a set of angles, I'm looking for a differentiable function with some specific properties; I'm ...
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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 ...
### Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time?
Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time? I'm trying to find the upper bound for the following series s(n) = 2^1 + 2^2 + 2^4 + 2^8 + 2^{16} + ...