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

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1answer
27 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
47 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
38 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
60 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
80 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
59 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
91 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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1answer
52 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} ...
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0answers
40 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.
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1answer
29 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
17 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
42 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
40 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
29 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
138 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
185 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
77 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
189 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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3answers
86 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 $\dfrac{\infty}{\infty}$, ...
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3answers
72 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 $x=0$.
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1answer
61 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 ...
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2answers
43 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? ...
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1answer
35 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
115 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} ...
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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 ...
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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 ...
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2answers
208 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 ...
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1answer
44 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
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1answer
91 views

Discrete Gaussian density function sum drawn from Gaussian distribution

Doing some analysis of my problem, I have come up with the following equation (I tried to search this problem but no luck) $S_N = \sum \limits_{i=1}^{N} \exp{\left(-\frac{X_i^2}{2\sigma^2}\right)}$ ...
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1answer
168 views

Exponential sum identity

How do I show that $$\sum_{|j| \leq J} (J-|j|) e(j \alpha) = \left| \sum_{j=1}^J e(j \alpha) \right|^2,$$ where $e(n)=e^{2 \pi i n}$ and $\alpha \not \in \mathbb{Z}$? Thank you very much in advance! ...
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1answer
137 views

Summation series for a $\times$ b^x

I know theres a summation series for basic $a^x$. Like $2^{x-1}$ is summed by $2^x-1$. Then how would you sum $5\times2^{x-1}$?
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1answer
129 views

summation of an arbitrary function multiplied by exponential

I'm trying to find some function $g(k)$ such that $$\sum_{k=0}^{\infty} g(k) \frac{(n \lambda)^k}{k!} = 0 $$ The textbook says that there is only one solution, that is $g(k)=0$ for all $k$. But I ...
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0answers
15 views

Multiple Integral Approximation

Does anybody know how to simplify following multiple integration ? ...
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0answers
29 views

Approximate Sum of exponentials

I want to closely approximate the following sum into possibly a single term (not as infinite summation) $SUM= \exp^{(-a_1x_1-a_2x_2-a_3x_3)} - \exp^{(-v)}\exp^{(-b_1x_1-b_2x_2-b_3x_3)}$ here $a_i, ...
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0answers
105 views

Derivative of Log of Summation of exponential function (base e)

A financial formula that I am implementing requires that I find the first derivative of a function to find a local maxima, from scratch. Can someone please help me with finding the first derivative of ...
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0answers
29 views

Maths for a half-life style calculation

Struggling to see a very simple way to do the following calculation. It strikes me that it must be similar to working out what percentage of uranium atoms in a sample of uranium will decay in a given ...
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1answer
52 views

Understanding solution to: $\sum_{k=0}^{N-1}\sum_{m=0}^{N-1}x[m]e^{-\rm j\frac{2\pi}{N}mk}e^{-\rm j\frac{2\pi}{N}nk}$

I cannot understand the solution that is presented below: \begin{eqnarray} X[k]&=&\sum_{m=0}^{N-1}x[m]e^{-\rm j\frac{2\pi}{N}mk}.\nonumber\\ y[n]&=&\sum_{k=0}^{N-1}X[k]e^{-\rm ...
0
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0answers
35 views

Simplified formula for the following sum

I have the following sum $$\sum_{n\ge 0}\sum_{0\le \alpha_i\le n+1,\\ \sum_{k=1}^N \alpha_i=n+1} \frac{e^{-(\rho+s)\left(\sum_{k=1}^N \alpha_kT_k\right)}\left(\rho\sum_{k=1}^N ...
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2answers
39 views

Inequation of an sum smaller than 1

I'm trying to figure out the following $$ \sum^{\infty}_{n=3} \dfrac{q!^2}{n!^2} < 1 $$ How I can show it if $q \geq 2$? Maybe with telescoping sums? Thanks, Landau
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0answers
43 views

Removing backups in an exponential fashion

Background: I want to create a backup system that utilizes the full space of a hard-disk. Given that all backups are approximately equal in size this means that I can save a fixed amount of backups. ...
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0answers
13 views

Find the inverse function of a function relating to limited exponential sum

The function is given out as: $$y = 4x + {x^m} + {x^{ - m}},where{\text{ 0 < m}} \leqslant {\text{1, 0 < }}y < 6;$$ Closed form will be highly appreciate,but approximate results is also ...
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0answers
11 views

Is there a 'mild' product function?

I'm simulating an economy, each person has a list of integers representing the quantity of each resource they possess (for example: 5 water, 6 food, 2 education). From this I want to calculate ...
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0answers
29 views

About complex exponential summation

Let $f:\,\mathbb{R}^{+}\rightarrow\mathbb{R},\, f\in C^{\infty}\left(\mathbb{R}^{+}\right)$ and such that $f\left(n\right)>0\,\forall n\in\mathbb{N}$. Let $c>0$ a real number, $N>0$ a large ...
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1answer
34 views

Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$. Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n ...
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1answer
42 views

Calculate remaining quantity based on half-life with everyday supplementation

Half life of a thing is 'h' days. I have a box where everyday I put 'n' quantity into the box. How do I calculate the quantity remaining after 'd' days. i.e. if half life is 7 days. Each day I add 10 ...
0
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1answer
88 views

exponential upper bound on sum of exponentials

For what value of $c_3$ can I guarantee that $(a+b)\exp(-c_3\theta)>a\exp(-c_1\theta)+b\exp(-c_2\theta)$ where $a,b,c_i>0$
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1answer
94 views

Finding a general coefficient in the multiplication of the two series

Help me please to find a general coefficient $a_j$ of the following series $$ ...
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3answers
39 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
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1answer
96 views

Equation to the summation to $ 3\cdot 10^{-n-1}$

What would the equation to sum (shown below) of $ 3\cdot 10^{-n-1}$? Just like $ 2^x-1$ is the sum of $ 2^{x-1}$. $$\sum_{n=0}^{\infty} 3\cdot 10^{-n-1}$$ This would be 0.3+0.03+0.003. . . this ...
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2answers
79 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}}- ...