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

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
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0answers
107 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
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2answers
41 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
40 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 ...
2
votes
0answers
97 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 ...
2
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0answers
80 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,\ ...
2
votes
1answer
96 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
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0answers
47 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
149 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 ...
1
vote
2answers
242 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: $10^{82}$ $1\times 2 = 2$ ...
1
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3answers
430 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
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2answers
65 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
47 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
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2answers
54 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? ...
1
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2answers
101 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
195 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 ...
1
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1answer
590 views

Use Cauchy's Multiplication Theorem and the Binomial Theorem to prove $\exp(x+y)=\exp(x)\exp(y)$

I am to use Cauchy's Multiplication Theorem and the Binomial Theorem in order to prove $\exp(x+y)=\exp(x)\exp(y) $ but I have no idea where to begin. All I can think of doing is setting $\exp(x)$ ...
1
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1answer
78 views

The limit of a sum with elements from 2 sequences

Let's consider the sequence of real numbers $(a_{n}),n\geq1,a_{1}>0$ that satisfies the following recurrence: $$\frac{n(n+2)}{(n+1)}a_{n}-\frac{n^2-1}{n}a_{n+1}=n(n+1)a_{n}a_{n+1}$$ I'm supposed ...
1
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2answers
250 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
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3answers
84 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
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2answers
28 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
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1answer
410 views

Non trivial upper bound for an exponential sum

Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for $$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
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2answers
32 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
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3answers
221 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
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1answer
23 views

Let $X~Bin(n, \lambda/n)$, $\lambda >0$. Show that for fixed $k \geq 0$, $P(X=k)\equiv \frac{e^-\lambda)\lambda^k}{k!}$

Let $X - Bin(n, \lambda/n)$, $\lambda >0$. By using approximation $(1-\frac{x}{n})^n\approx e^{-x}$. Show that for fixed $k\geq 0$, $P(X=k)\approx \frac{e^{-\lambda}\lambda^k}{k!}$ ...
1
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1answer
69 views

Is there a closed form for $\sum_{n=1}^\infty \frac{q^{-n^2}}{n}$?

I found that $\sum_{n=1}^\infty q^{-n^2}$ can be expressed using a theta function. Is there also a closed form for the following series? $$\sum_{n=1}^\infty \frac{q^{-n^2}}{n}$$
1
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1answer
24 views

Trigonometical sum from Fourier analysis

(Edit) Note: $a\in \mathbb{R},0<a\leq\pi$. Also, the sum skips $n=0$ (that's where the other term comes from). Working through a Fourier analysis exercise I've got stuck in a clearly ...
1
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1answer
104 views

Solve equation with unknown in exponents

This is in continuation of this but not related to it completely. I am interested in finding a solution to the equation: $m' = m - \sum \limits_{j=1}^{m} (1 - d_{O_j}/n)^k$. where $m,m',n$ and ...
1
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1answer
61 views

Help in simplifying this nasty expression obtained after binomial expnasion

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer, $$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} ...
1
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2answers
60 views

The result of exponential sum formula

I am awakard to deal with math problem I am trying to understand the first condition that is (k-r)=mN I can understand when (k-r) is mN, the left formula is 1 But I don't know how to left ...
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2answers
42 views

Identity involving trigonometric sum

I have to prove that $$\overset{N}{\underset{n=-N}{\sum}} \left(N-\left|n\right|\right)e^{2\pi inx}=\left|\overset{N}{\underset{n=1}{\sum}} e^{2\pi inx}\right|^{2}=\left(\frac{\sin\left(N\pi ...
1
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1answer
75 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
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4answers
88 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
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2answers
104 views

How to approximate this large sum of exponential terms

Is there any way to approximate the following sum: $$ \sum\limits_{i_1=1}^N\sum\limits_{i_1=2}^N \cdots \sum\limits_{i_k=1}^N \cdots\sum\limits_{i_N=1}^N \exp(-r_{i_1}-r_{i_{k+1}}-r_{i_{2k+1}}- ...
1
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1answer
68 views

Complex exponential sumation: $\sum_{-M}^M e^{-2\pi ikf}$

Does someone know how to prove that $\displaystyle {\sum_{k = -M}^M e^{-2\pi ikf} = \frac{sin((2M + 1)\pi f)}{sin(2\pi f)}}$ where $f$ is a constant such that $-\frac{1}{2} < f < \frac{1}{2}$ ...
1
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1answer
96 views

What is the $\sum\limits_{i=0}^{\ (\log_2(n))-1)}\frac{n}{2^i}$?

What is the value of the following sum: $$\sum\limits_{i=0}^{\ (\log_2(n))-1)}\frac{n}{2^i}$$ Can you show how to go about arriving at the answer?
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0answers
144 views

sum of exponential series with power increasing by geometric series

Is there any way to reduce the following summation... $2^{ar^0}+2^{ar^1}....+ 2^{ar^n} $ to a simple equation? I feel like I can pull out a $2^a$ somehow and then treat it as a normal series but I ...
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0answers
33 views

Plotting exponential partial sums in the complex plane

I was plotting the following sequence of points $(a_n)_{n = 0}^\infty$ in the complex plane for various reals $\alpha > 1$: $$a_n = \sum_{k = 0}^n e^{i k^\alpha}$$ I found that for many values of ...
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0answers
19 views

Solve $b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x$

Suppose, \begin{align*} b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x \end{align*} Assume $a_1,a_2,a_3, b_1,b_2, b_3>0$ What are the possible values of $a_1,a_2,a_3, b_1,b_2, ...
1
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1answer
33 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 ...
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0answers
24 views

Help in deriving proof with exponential and little-oh

I'm working through a proof of the Kelly Criterion found at http://www.elem.com/~btilly/kelly-criterion/. In checking up on some of the derivations I can't follow the last step in this sequence ...
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0answers
44 views

please evaluate this sum

$$\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{3^{a+n}}{a+n}-\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{1}{a+n}$$ I need help evaluating the above, step by step would help. Thanks for a>0 or ...
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1answer
43 views

Is it possible to solve this nonlinear equation analytically?

Is it possible to solve the following equation analytically? $B_1\exp(\beta_1 x) + B_2\exp(\beta_2 x) = C_1\exp(\alpha_1 x) + C_2\exp(\alpha_2 x)$ where, $B_1$, $B_2$, $C_1$, $C_2$, $\beta_1$, ...
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0answers
35 views

Math equations of electron scattering

I'm trying to figure out the missing step here, in a problem about X-ray crystallography. I am referring to the attached image: In the image, A= electron density, Z= distance traveled, λ= X-ray ...
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0answers
166 views

sum of $p$th roots of unity

Let $p$ be an odd prime. Let $\mathbb Z_p$ be the ring of integers from $0$ to $p-1$, i.e. $\mathbb Z_p=\left\{0,1,\dots,p-1\right\}$. Let $r$ be a positive integer. Then, for all values of $p$, is ...
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0answers
272 views

Approximation of an exponential sum

Consider the flowing exponential sum with $x,\,y \ge 0$ and $c_i,\, d_i$ is a real number for $i=1,..,N$ $$ E=\sum\limits_{i = 1}^N \exp \left(- \left( x - c_i \right)^2 - \left( y - d_i \right)^2 ...
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0answers
85 views

Sum of power and Bernoulli intuitive discover

I really looked up to the 10th page of google, and the PDF's I find aren't complete, and definetively have NOT intuitive explanation about Bernoulli's discover. I know that he observed the formulas of ...
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0answers
196 views

Non-trivial upper bound for binomial sum

I'm trying to upper bound the following sum: $$ \sum\limits_{k=0}^n \begin{pmatrix} n \\ k \end{pmatrix} e^{-\frac{m}{2^k} } $$ where $n>0$ is fixed and $0\leq m \leq 2^n$. A trivial upper ...
0
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3answers
75 views

$\sum\limits_{n=4}^{\infty } \frac{2^n + 8^n}{10^n} = ?$

im looking for hints on how to do: $\sum\limits_{n=4}^{n= \infty } \frac{2^n + 8^n}{10^n} = ?$ I thought this may have had something to do with geometric series but nothing obvious comes up ...
0
votes
3answers
91 views

Can I know the value of an infinite serie?

$$\sum\limits_{n=0}^{\infty}\frac{n}{e^n}$$ I have found through a software that the value is $\dfrac{e}{(e-1)^2}$. I've been trying to do it manually but I am getting $\dfrac{\infty}{\infty}$, ...