# Tagged Questions

For questions on exponential sums.

4answers
118 views

### Obtain $x$ from exponential equation

I'm looking for $x$ from the following equation: $$\frac1{(1+x)^{\frac1{p}}}+\frac1{(1+x)^{\frac2{p}}} + \cdots \frac1{(1+x)^{\frac{n}{p}}} = 0$$ Or, what's the same: ...
2answers
32 views

### What's the radius of convergence of the next sum: $\sum_{n=0}^\infty (\int_o^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt)x^n$

What's the radius of convergence of the next sum: $$\sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n$$ I know that $$\int_0^\infty\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt$$ does ...
0answers
56 views

### Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+1-k} ...
1answer
64 views

### Exponential Sum - Solve for X

I'm wondering if there is a way to solve for x given the following equation: $$A^x + B^x = C^x$$ where A, B, and C are known constants. For pythagorean triples, $x = 2$. I've seen a lot of stuff for ...
2answers
56 views

### Inequality with a sum

I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos http://arxiv.org/abs/1107.5301 I am stuck at inequality in proof of Lemma 6. $n\geq 8$, \$k=2\lfloor ...
0answers
28 views

### Normalizing sum of radial basis functions

I am trying to simulate decision making using a sum of 3-dimensional radial basis functions. I have $n\in [1,\infty]$ radial basis functions like \$f_i(x,y,z) = A_i\ ...
1answer
50 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! ...
1answer
64 views

### Exponential as power series

Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$? Just to be clear, the summation starting from 1 is intentional, otherwise the ...
0answers
48 views

### Inequality : Partial Sum of $N$th root of unities

Let $\hat{\lambda}_{B}(l)=\sum_{n=0}^{B-1}e^{j\frac{2\pi}{N}nl}$, where $N$ is any positive integer and $1\leq B\leq N$ and $0 \leq l \leq N-1$. $\hat{\lambda}_{B}(l)$ is the partial sum of the ...
0answers
36 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 ...
4answers
120 views

### Summation Simplification

I am attempting to solve a problem that I posed myself, but I can't figure out how to simplify the solution from the "messy" state in which it currently exists. My mathematical background does not yet ...
1answer
182 views

### sum of $\displaystyle \frac{\sin nx }{n^4}$

Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$ Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
3answers
38 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 ...
1answer
86 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} ...
3answers
63 views

### How can we represent any number as a series of exponents?

Say I have a positive integer that is one-thousand digits long. What math could I use to represent this number as a series of exponents in a significantly shorter form than the original number? The ...
1answer
159 views

### Proof for generalized sum of powers

Bernouli's Formula for sum of kth powers of first n natural numbers is given by: $$f_k(n)=\frac{1}{k+1}\sum_{j=0}^k{k+1\choose j}B_j(n+1)^{k+1-j}$$ where $Bj$ is the $j^{th}$ Bernoulli Number and is ...
1answer
167 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 $\exp(x)$ ...
0answers
194 views

### upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
1answer
61 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 $5\times2^{x-1}$?
0answers
27 views

### Summation of $3 \times 10^{-n-1}$ [duplicate]

Possible Duplicate: 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 $2^{x-1}$. ...
1answer
83 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 ...
1answer
213 views

### How to find a summation of a logarithmic function?

Suppose that I had to find $\log_{10}(8952!)$. Now, since $\log(a) + \log(b) = \log(ab)$, this can be rewritten to the following summation: $$\sum_{x=1}^{8952}{\log_{10}(x)}$$ Would there be a ...
3answers
112 views

### Sum of the sequence

What is the sum of the following sequence \begin{align*} (2^1 - 1) &+ \Big((2^1 - 1) + (2^2 - 1)\Big)\\ &+ \Big((2^1 - 1) + (2^2 - 1) + (2^3 - 1) \Big)+\ldots\\ &+\Big( (2^1 - 1)+(2^2 - ...
4answers
151 views

### Help evaluating a limit

I have the following limit: $$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$ where $\alpha>0$. ...
1answer
156 views

### sum of infinite series

Does there exist an explicit expression for $$\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{k}e^{-\lambda \left( 2k+1\right) ^{2}}}{\left( 2k+1\right) ^{3}}\;,$$ where $\lambda$ is a positive scalar? ...
1answer
80 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  ...
2answers
106 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: $10^{82}$ $1\times 2 = 2$ ...
3answers
249 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 ...
1answer
49 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 ...
1answer
70 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?
1answer
68 views

### Two questions on trigonommetric sums and integrals

Is it true that $\int_{0}^{\infty}\sin(mx)\sin(nx) \, dx = \delta (m-n)$ although using Euler formula I get a linear combination of $\delta(m-n)$ and $\delta (m+n)$? What is the sum ...
1answer
1k 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 ...
2answers
80 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 ...
1answer
34 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 ...
0answers
147 views

### Equidistribution of roots of prime cyclotomic polynomials to prime moduli

Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
0answers
147 views

### Complexity of TSP using Dynamic Programming [closed]

I was reading about TSP using Dynamic Programming and the complexity was given as $\theta (n^2 2^n)$ Certain intermediate calculations are given as \$N= \Sigma_{k=0}^{n-2}(n-1)C(n-2,k) = ...
0answers
123 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 ...
1answer
141 views

### Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right)$$ for general $k$ and $n$? For $k=1$, on MathWorld we have that ...
2answers
122 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 ...
1answer
110 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 ...
1answer
214 views

### Non trivial upper bound for an exponential sum

Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for $$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 77, "mathjax_display_tex": 19, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9099400043487549, "perplexity_flag": "middle"}