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

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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
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3answers
147 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
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2answers
224 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
418 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 ...
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2answers
60 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 ...
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2answers
42 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?
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2answers
53 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? ...
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2answers
96 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}, ...
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1answer
194 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
555 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 $\exp(x)$ ...
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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 ...
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2answers
238 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 ...
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2answers
27 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
392 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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3answers
169 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.
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2answers
18 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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1answer
66 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
23 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
50 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 ...
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2answers
54 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
40 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 ...
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1answer
69 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
86 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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1answer
61 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 $-\frac{1}{2} < f < \frac{1}{2}$ ...
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1answer
95 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
48 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
25 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, ...
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1answer
56 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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1answer
32 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
21 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
43 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
41 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, $B_1$, $B_2$, $C_1$, $C_2$, $\beta_1$, ...
1
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0answers
33 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
162 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
242 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
79 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
194 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 ...
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2answers
63 views

Exponential sum [duplicate]

$$ \sum_{k=1}^\infty \frac{k^2}{2^k} = 6$$ But how? What formula should we use? It is not geometric series. Thanks Ann
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3answers
89 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 $\dfrac{\infty}{\infty}$, ...
0
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3answers
73 views

How would you solve for $x$ in $7^{x}=5^{x-4}$?

How would you be able to make these the same base? I tried to put it into a base $10$ formula, but it ended up making $x=0$.
0
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1answer
166 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 ...
0
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2answers
66 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
0
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1answer
36 views

Arrangement of the following term

How $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2m}}$$ can be rewritten as $$\sum_{k=1}^{\infty}\frac{(-1)^{2k-2}}{(2k-1)^{2m}}+\sum_{k=1}^{\infty}\frac{(-1)^{2k-1}}{(2k)^{2m}}$$ ???
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1answer
57 views

quadratic Gauss sum over a power of 2

Is there a general formula for the generalized quadratic Gauss sum defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$ and $c$ is a power of ...
0
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1answer
119 views

A hard limit with integral sign

$$\displaystyle\underset{m\to +\infty }{\mathop{\lim }}\,\left[ \underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\int_{0}^{x}{{{\left( \underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1-\frac{1}{n} ...
0
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1answer
3k 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 ...
0
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1answer
35 views

Specific range of numbers is given, trying to get another number within same range

I'm trying to calculate the width of an HTML element based on the window size. Here's what I have. These width values (first value) accurately match with the width the HTML element must be (second ...
0
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2answers
218 views

Exponential formula to assign list items an equally distributed percentage

I am going to explain this as best as I can: We have a number of list items, that dynamically changes (sometimes it has 5 items sometimes it has 7 items or 43 items, etc.) We are trying to find a ...
0
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2answers
42 views

Get $a_1$, $a_2$, $b_1$, $b_2$ from $a_1 \times \exp{b_1 \times x} - a_2 \times \exp{b_2 \times x}$

I have experimental data which follow the function below. $$f(x) = a_1 e^{-b_1 x} - a_2 e^{-b_2 x} + \epsilon$$ ($a_1$, $b_1$, $a_2$, $b_1$ are all positive real numbers. $\epsilon$ represents ...