# Tagged Questions

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

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### Convergence of $\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$

We know that $\sum_{ k\in\mathbb{N} } \frac{\lambda^k}{k!} = e^\lambda$. I'm interested in the convergence of $$S^{(n)}=\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$$ for some value ...
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### Characters on rings of residue classes modulo polynomials over finite fields

First recall the following orthogonality relation on $\mathbb{Z}/n\mathbb{Z}$. Fix $n \in \mathbb{Z}$, $n \neq 0$. For $r \in \mathbb{Q}$, let $e(r) := e^{2 \pi i r}$. Let $x \in \mathbb{Z}$. Then ...
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### Standard deviation of phase for a random phasor sum

I have a phasor sum $a e^{j \theta} = \frac{1}{\sqrt{N}} \sum_{k=1}^{N} \alpha_k e^{j \phi_k }$ where $\phi_k = [-\pi, \pi]$, the standard deviation $\sigma_{\phi}$ of the phase is known and the ...
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### Exponent Subtraction With Same Base

Can someone explain how this is valid? Thanks so much! $3\cdot 2^{2n+2} - 3\cdot2^{2n} = 3\cdot2^{2n}$
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### Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
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### Sum of exponential functions involving powers of two

I came across a weird series with exponential functions and powers of two: $$\sum_{k=0}^{\infty} \left(1 - e^{-2^{-k}z} \right), z \in \mathbb R_+$$ and have no idea how to solve this (if there even ...
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### Equality of exponential functions from geometric series

I'm currently trying to understand why the first and second line of this equation are in fact equal. This is taken from "Introduction to the Physics of Waves" by Tim Freegarde from a chapter about ...
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### What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it? EDIT: Apologies. I can see this converges by the ratio test. My issue is working out its sum, more for fun really. It ...
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### Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
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### Median and Mean of Sum of Two Exponentials

I have a cumulative distribution function: $$G(x) = -ae^{-xb} - ce^{-xd}+h$$ The associated probability density function is: $$g(x) = abe^{-xb} + cde^{-xd}$$ My problem concerns $x\ge 0, X \in R$. I ...
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### Integrating exponential of multiple exponentials

I have a integral term that looks similar to $\int_0^\infty\exp(-u-ae^{-c_1u}-be^{-c_2u})\,du$ where the constants $a,b,c_1,c_2>0$. For the case where $b=0$ I can use the answer from: Integrating ...
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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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### 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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### Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
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### Does this series converge? If so, to what?

i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am ...
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### The identity of a polynomial sum

I am wondering if there is a recursive formula to calculate $$S=1^{k}+2^{k}+3^{k}+\dots+n^{k}$$ Where $k$ and $n$ are natural numbers.
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### Summing a exponential series

What is the appropriate way to simplify such an expression. i am unsure of how to use the series i know to apply to this situation $$\sum_{L=0}^{M}s^{L}L^{2}$$ do i modify such a series as power ...
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### Reducing infinite summation

I'm trying to reduce the following nested summation (removing all summations), using the fact that: $\sum_{i=0}^\infty\ a^i = 1/(1- a)$. Problem: $\sum_{n=0}^\infty\sum_{m=n}^\infty\ a^nb^m$ I know ...
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### Solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$

How can I solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$, where $N\geq 1, k_1,\ldots,k_N \in \mathbb{R}, k_1,\ldots,k_N < 0, x\in \mathbb{R}$ and $x >0$. I looked at the basic rules of ...
### How do I solve for $\delta$ in this question
$316.45 = 100e^{\delta(10)} + 100e^{\delta(5)}$ I don't know why I can't do this. I thought of using $\ln$ but I don't think $\ln(A+B) = \ln(A) + \ln(B)$ or does it?