# Tagged Questions

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

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### Sum of a series with exponential and polynomial terms

I have reduced the expression that I am working on to the following sum of series, which is definitely converging. It would be great if someone can help me out with this or suggest ways this can be ...
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### Does this sequence have a closed form representation?

We know that $$\sum_{s=0}^\infty \frac{\lambda^{s}}{s!} = e^\lambda$$ Relatedly, $$\sum_{s=1}^\infty \frac{\lambda^{s}}{s!}s = \lambda \sum_{s=1}^\infty \frac{\lambda^{s-1}}{(s-1)!}$$ For which ...
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### Infinite exponential sum doubt

Hello! I have a couple of doubts regarding a formula seen here : $$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$ What would happen if the real part of z Re(z) were equal ...
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### Calculation of an integral involving the sum of a range of natural exponential functions

Does somebody know how to solve the following integral, I extremely hope I can obtain its close-form solution: \int \sqrt{ \sum_{i=1}^{M}\sum_{j=1}^{M} e^{-\frac{\frac{\left|\mathbf{...
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### The upper bound of a sum of exponential function

Could someone help me to find the upper bound of the following function: $f(x) = \sqrt{\sum_{n=i}^{N} e^{-\alpha_{i}\cdot x}}$, where $x > 0$, the $i^{th}$ coefficient $\alpha_{i} > 0$. I got ...
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### Solving an equation involving the trace of a field

Let $F$ be a finite field of order $q$ where $q=2^{n}$ and fix $l\in F\setminus{0}$ with $Tr(l)=0$. I want to determine the number of $a$ such that $$Tr(la)=Tr(la^{-1})=1,$$ where $Tr$ denotes the ...
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### series summation of quadratic exponentials

I need to prove the equation $\sum_{n=-\infty}^{\infty} e^{-\frac{1}{2} An^2+iBn} = \sqrt{\frac{2\pi}{A}} \sum_{l=-\infty}^{\infty} e^{-\frac{1}{2A}(B-2\pi l)^2}$, where A and B are constants(possibly ...
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### Convergence of a convergence series with $e^{in}$

I am facing difficulties with this question: It says show by using the comparison test that the folowing complex series converges: $$\sum_{n=1}^\infty \frac{\Re(e^{in\phi})}{2^n}$$ The $\Re$ refers ...
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### How to solve $6^{x} + 8^{x} + 9^{x} = 12^{x}$?

I checked on WolframAlpha and it says the answer is approx. 2.56639, but I don't know how to go about solving this equation. What kind of approach should I use?
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### best values in the estimate of Vinogradov-Korobov

Let $C(N)=\sum_{1<n\le N}{n^{-it}}$. Vinogradov- Korobov estimate is $$|C(N)| \le KN\exp\left(-\gamma \frac{\ln^3 N}{\ln^2 t}\right).$$ What are the best values of K and $\gamma$ ? I have only ...
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### upper bound formula the binomial coefficients with real valued arguments

I'm trying to prove the following. Let $n\in\mathbb{N},m\in\mathbb{N}\cup\{0\},\alpha\in (n-1,n)$ and $N\in\mathbb{N}:N\ge m+1$. Prove that \begin{align} &\sum_{k=N+1}^\infty\Big{|}\binom{n+m-\...
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### Generate exponential weights (sum of all = 1)

I have $500$ observations and I want to make exponential weighted average of them. I want the weights to be something like $w_i = 0.999^t$ when $t$ is from $1$ to $500$ (num of observations). ...
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### How to simplify a sum of exponential equation?

Suppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ...
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### Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no success....
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### log of summation expression

I am curious about simplifying the following expression: $\log \left(\sum_\limits{i=0}^{n}x_i \right)$ Is there any rule to simplify a summation inside the log?
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### How to simplify a sum for the total cost of a yearly payment including compound interest

I want to simplify the below sum for the total cost over a yearly payment including compound interest over n years. An example: we have 150 euros that need to be paid every year and an interest of 2%....
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### Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
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