For questions on exponential sums.
0
votes
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:
...
2
votes
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 ...
1
vote
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
$$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+1-k} ...
3
votes
1answer
64 views
Exponential Sum - Solve for X
I'm wondering if there is a way to solve for x given the following equation:
$$A^x + B^x = C^x$$
where A, B, and C are known constants. For pythagorean triples, $x = 2$. I've seen a lot of stuff for ...
3
votes
2answers
56 views
Inequality with a sum
I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos
http://arxiv.org/abs/1107.5301
I am stuck at inequality in proof of Lemma 6.
$n\geq 8$, $k=2\lfloor ...
0
votes
0answers
28 views
Normalizing sum of radial basis functions
I am trying to simulate decision making using a sum of 3-dimensional radial basis functions.
I have $n\in [1,\infty]$ radial basis functions like
$f_i(x,y,z) = A_i\ ...
0
votes
1answer
50 views
Exponential sum identity
How do I show that $$\sum_{|j| \leq J} (J-|j|) e(j \alpha) = \left| \sum_{j=1}^J e(j \alpha) \right|^2,$$
where $e(n)=e^{2 \pi i n}$ and $\alpha \not \in \mathbb{Z}$?
Thank you very much in advance!
...
1
vote
1answer
64 views
Exponential as power series
Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$?
Just to be clear, the summation starting from 1 is intentional, otherwise the ...
0
votes
0answers
48 views
Inequality : Partial Sum of $N$th root of unities
Let $\hat{\lambda}_{B}(l)=\sum_{n=0}^{B-1}e^{j\frac{2\pi}{N}nl}$, where $N$ is any positive integer and $1\leq B\leq N$ and $0 \leq l \leq N-1$. $\hat{\lambda}_{B}(l)$ is the partial sum of the ...
1
vote
0answers
36 views
a functional space for the study of exponential sum
I would like to study the exponential sum in an appropriate functional space.
In particular:
$f(x): \mathbb{R} \to \mathbb{R} $
$f(x) = \chi_K \sum_{i=1}^{M} R_i \exp{(s_i x)}$
where $R_i, s_i \in ...
4
votes
4answers
120 views
Summation Simplification
I am attempting to solve a problem that I posed myself, but I can't figure out how to simplify the solution from the "messy" state in which it currently exists. My mathematical background does not yet ...
4
votes
1answer
182 views
sum of $\displaystyle \frac{\sin nx }{n^4}$
Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$
Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
1
vote
3answers
38 views
Need to check simplification of expression with infinite sum of exponentials
In reviewing a paper, I've come across a simplification the looks fishy to me, but I'm having a hard time checking it. I pulled out my old CRC handbook, but neither that nor Google are proving to be ...
0
votes
1answer
86 views
A hard limit with integral sign
$$\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} ...
2
votes
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 ...
3
votes
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 ...
1
vote
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 $\exp(x)$ ...
1
vote
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 ...
0
votes
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 $5\times2^{x-1}$?
0
votes
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}$.
...
-1
votes
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 ...
3
votes
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 ...
4
votes
3answers
112 views
Sum of the sequence
What is the sum of the following sequence
$$\begin{align*}
(2^1 - 1) &+ \Big((2^1 - 1) + (2^2 - 1)\Big)\\
&+ \Big((2^1 - 1) + (2^2 - 1) + (2^3 - 1) \Big)+\ldots\\
&+\Big( (2^1 - 1)+(2^2 - ...
6
votes
4answers
151 views
Help evaluating a limit
I have the following limit:
$$\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!}$$
where $\alpha>0$.
...
6
votes
1answer
156 views
sum of infinite series
Does there exist an explicit expression for
$$\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{k}e^{-\lambda \left(
2k+1\right) ^{2}}}{\left( 2k+1\right) ^{3}}\;,$$
where $\lambda$ is a positive scalar? ...
0
votes
1answer
80 views
Finding a general coefficient in the multiplication of the two series
Help me please to find a general coefficient $a_j$ of the following series
$$
...
1
vote
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: $10^{82}$
$1\times 2 = 2$
...
1
vote
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 ...
1
vote
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 ...
1
vote
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?
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
7
votes
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
$E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
0
votes
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) = ...
1
vote
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 ...
4
votes
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
...
0
votes
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 ...
0
votes
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 ...
1
vote
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
$$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
