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

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2
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1answer
70 views

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 ...
2
votes
1answer
56 views

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 ...
2
votes
1answer
176 views

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 ...
2
votes
1answer
28 views

Exponent Subtraction With Same Base

Can someone explain how this is valid? Thanks so much! $3\cdot 2^{2n+2} - 3\cdot2^{2n} = 3\cdot2^{2n}$
2
votes
1answer
60 views

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 $F$ ...
2
votes
1answer
177 views

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 ...
2
votes
1answer
132 views

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 ...
2
votes
1answer
155 views

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 ...
2
votes
4answers
67 views

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 ...
2
votes
1answer
146 views

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 ...
2
votes
1answer
187 views

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 ...
2
votes
1answer
58 views

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 ...
2
votes
1answer
186 views

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-\...
2
votes
0answers
51 views

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 $\phi \colon R \times R$ ...
2
votes
0answers
52 views

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 ...
2
votes
2answers
171 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where $\...
2
votes
0answers
47 views

an exponential sum involving quadratics

Let $q$ be a prime and $a,b\in \mathbb{Z}$. Can we compute $$\sum_{k=1}^{q}e^{2\pi i\frac{ak^2+bk}{q}}$$ in terms of $a,b,q$? The sum is easy when $a=0$, but what about the case $a\neq 0$?
2
votes
1answer
98 views

Bounds on sum of cosines

Find bound on the following sums for $k\in \{1,\dots p-1\}$ where $p$ is a prime (even assumed to be $3\mod 4$, I used this to get to the current sum) find good upper and lower bounds on $$\sum_{s=0}...
2
votes
1answer
80 views

Express sum of complex exponentials as 1 + sum of cosines

I'm told that I can express the following sum of complex exponentials: $$\sum_{n=0}^6 e^{-j\Omega n}$$ As 1 plus 3 cosine terms. I'm having a really hard time arriving at this, I see where the 1 ...
2
votes
0answers
270 views

The sum of finite exponential series with a quadratic phase

How can I prove that: $$ \sqrt \frac K2 + i \sqrt \frac K2=\sum^K_{m=1}\exp\left(i\frac \pi Km^2\right) $$ When $K$ is even.
2
votes
1answer
35 views

Proof that $ y(n) = ∑_{k=-∞}^{∞}\ {a}^{-k}u(n-k)u(-k) = \frac{1}{1-a }$ if $n>0$

Can someone explain the steps and how the boundaries for the summation change to result in the answer (And possible for the case where $n\leq 0$. I am not really a mathematician, don't know if the ...
2
votes
2answers
47 views

Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $ 3^{x+2} + 2^x = 5 $ The teacher said this equations can be solved ...
2
votes
1answer
46 views

Evaluation of infinite sum related to problems in elasticity

I'm working on some problems with relation to elasticity (plate mechanics in specific) and while I've made some progress the following sum is giving me a hard time $\sum_{m=0}^{\infty}\frac{(-1)^m}{(...
2
votes
0answers
106 views

Descartes Rule of Sign for exponential sums

I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like $$f(x)=a_1x^3b_1^x+a_2x^2b_2^x+...
2
votes
0answers
81 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ 0\...
2
votes
1answer
100 views

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.
2
votes
0answers
53 views

a functional space for the study of exponential sum

I would like to study the exponential sum in an appropriate functional space. In particular: $f(x): \mathbb{R} \to \mathbb{R} $ $f(x) = \chi_K \sum_{i=1}^{M} R_i \exp{(s_i x)}$ where $R_i, s_i \in ...
2
votes
3answers
155 views

Need to check simplification of expression with infinite sum of exponentials

In reviewing a paper, I've come across a simplification the looks fishy to me, but I'm having a hard time checking it. I pulled out my old CRC handbook, but neither that nor Google are proving to be ...
2
votes
2answers
50 views

sum of fractional powers

Let $z=e^{i\theta}$ for some real number $\theta$, i.e. $z$ is complex number on the unit circle. Is there a formula for $z+z^{1/2}+z^{1/2^2}+\dots+z^{1/2^n}$? Here $n$ is a positive integer. I only ...
1
vote
2answers
443 views

How many times do you need to double previous result to get at least $10^{82}$?

This is pretty straightforward, but I'd like to study, how find out, how many times do you need to double previous result of calculation to get some sum, for example: $10^{82}$ $1\times 2 = 2$ $2\...
1
vote
3answers
450 views

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 ...
1
vote
3answers
100 views

Prove $\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $ [duplicate]

I have to prove that $$\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $$ Such that ${n \choose i} = \frac{n!}{i!(n-i)!} $ and $n $ is some arbitrary int I proved we can expand 2^I in a way such that $...
1
vote
2answers
91 views

What is the distribution of the average of IID exponential RVs? (have to use MGFs)

Just a quick question as I cannot figure this out. Here is the problem: Q: What is the distribution of the following? ($Y_1 + Y_2 + Y_3 +\cdots+ Y_n$) $ /n$ if each $Y$ is independent and identically ...
1
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2answers
53 views

Question on power, If 2x^2x^2x^2x… =4 Solve for x

I've seen this random example, in which can anyone give me clue how to solve for $ x $ here?
1
vote
2answers
57 views

Summation simplification on exponentiaion

I would like to ask help for the following question: Simplify $$\sum_{x=0}^{+\infty}{e^{-x/\mu}}$$ Will integration work here? Meaning, can I do it this way? $$\sum_{x=0}^{+\infty}{e^{-x/\mu}}=\...
1
vote
1answer
729 views

Stuck on derivative of logarithm of sum of exponentials

let's say that I need to calculate the following expression: $$ \frac{\partial\mathrm{log}(\mathrm{exp}(w_1 * x_1 + b_1) + \mathrm{exp}(w_2 * x_2 + b_2))}{\partial w_1} $$ How do I start? The rules ...
1
vote
2answers
129 views

How to solve these series?

Can anyone help me understand how to solve these two series? More than the solution I'm interested in understanding which process I should follow. Series 1: $$ \sum_{i = 3}^{\infty} i * a^{i-1}, ...
1
vote
1answer
229 views

Summation of roots of unity equal to zero

Let $u$ be a integer which has an odd prime $m$ as a divisor. So we let $u=u_1m$ where $u_1=\frac{u}{m}$. Consider the set of $u$th roots of unity, i.e., $A=\{e^{j\frac{2\pi}{u}n} | 0 \leq n \leq u-...
1
vote
2answers
326 views

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 ...
1
vote
3answers
91 views

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 ...
1
vote
2answers
29 views

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?
1
vote
1answer
25 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= \sum_{\alpha<\beta}K_\...
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2answers
93 views

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 ...
1
vote
2answers
67 views

Limit log-sum of exponentials

I'm trying to compute the following limit: $$\lim_{\lambda \rightarrow \infty} \frac{1}{\lambda}\log\sum_{i=1}^n \exp[\lambda a_i]$$ I tried to solve with L'hoptials: $$= \lim_{\lambda \rightarrow ...
1
vote
3answers
528 views

Solve for Exponent in a summation

For a simple summation: $$ Z = \sum_{x=1}^X x^n $$ Solve for n. I have googled unsuccessfully and I do not remember this.
1
vote
1answer
5k views

Continuous Random Variable - Uniform Median, Exponential Mode

Working on this question: The median of a continuous random variable with CDF $F(x)$ is the value $m$ that guarantees that $$P\{X > m\} = P\{X < m\} = \frac{1}{2}$$ The mode is the ...
1
vote
1answer
30 views

Linear exponential sum over $(\mathbb{Z}/(p^t))^*$

Let $p$ be a prime and $t$ a natural number. Let us denote $(\mathbb{Z}/(p^t))^*$ to be the group of units of $\mathbb{Z}/(p^t)$. I have the following exponential sum $$ S = \sum_{w \in (\mathbb{Z}/(...
1
vote
1answer
27 views

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 ...
1
vote
1answer
28 views

How can we prove that the discrete Fourier transforms preserves the inner product up to a constant factor?

Let $d\in\mathbb N$, $\omega\in\mathbb C$ be a primitive $n$-th root of unity and $$\operatorname{DFT}_\omega:\mathbb C^d\to\mathbb C^d\;,\;\;\;z\mapsto\left(f_z\left(\omega^0\right),\ldots,f_z\left(\...
1
vote
2answers
30 views

Is $\int z^n e^{az}dz $ a combination of exponentials and polynomials?

We have $$I(n)=\int z^n e^{az}dz=\int z^n \left (\frac{1}{a}e^{az}\right )'dz=\frac{1}{a}z^ne^{az}-\frac{1}{a}\int nz^{n-1}e^{az}dz \\ \Rightarrow I(n)=\frac{1}{a}z^ne^{az}-\frac{1}{a}nI(n-1) \ \ \...