Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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1
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0answers
20 views

Evaluating a limit involving summation [duplicate]

Evaluate : $$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$ Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried ...
1
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0answers
20 views

Sum of Logarithms of expoenetial functions

Given a matrix $P \in \Re^{n \times d}$, and a column vector $\theta \in \Re^d$. Assume that $\sum\limits_{i=1}^n \ln{(1+e^{P_i\theta})} \leq 1$, where $P_i$ is the $i^{th}$ row in $P$. What can be ...
11
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1answer
86 views

Evaluating $\sum_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$

For $k=1$, the series does not converge. When $k=2$, I can prove that: $$\sum_{n=1}^\infty\frac{1}{n \binom{2n}{n}}=\frac{\pi}{3\sqrt{3}}$$ Using the result of: $$\int_0^\infty \frac{x^ndx}{(x+1)^{...
3
votes
2answers
47 views

Proving $\sum_{k=0}^n \binom n k (-2)^n =(-1)^n$

Can someone prove that $$\sum_{k=0}^n \binom n k (-2)^k =(-1)^n $$ ?
0
votes
1answer
28 views

How to simplify a sum involving the floor function

Let us suppose the positive integers $a$, $b$ and $n$ with $a<b$. Is it possible to simplify the following sum: $$2 \left\lfloor \frac{an}{b} \right\rfloor b + \frac{2^2}{3} \sum_{j=1}^{n-2} 3^j \...
1
vote
2answers
46 views

Prove the inequality between integral and summation of multiplicative inverse

I want to prove the following inequality: $$ \ln(n) = \int\limits_1^n{ \frac{1}{x} dx } \geq \sum_{x = 1}^{n}{\frac{1}{x + 1}} = \sum_{x = 1}^{n}{\frac{1}{x}} - 1 $$ I ask this question as I'm ...
8
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0answers
61 views

The pigeonhole principle - how to solve questions like that?

We have two sequences , $(a_i)_{i=1}^{2n}$ and $(b_i)_{i=1}^{2n}$ such that $1\leq a_i, b_i\leq n$ for every $i$. Show that there are two sets of indexes $I, J \subseteq \left \{ 1,2, ... 2n \right \...
5
votes
2answers
115 views

Are there $a,b \in \mathbb{N}$ that ${(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b $ beside $2,3$

We know that: $$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3 $$ My question is there other examples that satisfies: $$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b $$
1
vote
1answer
35 views

Finding the lower and upper sums of $f(x)=2x+1$

I'm trying to find the upper and lower sums for $f(x)=2x+1$ on the interval $[1,4]$. $P_6$ is the partition of $[1,4]$ consisting of 7 equally space points, $1, \frac{3}{2}, 2, \frac{5}{2},3,\frac{7}{...
0
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2answers
32 views

Is there a way to iterate through a set?

I have a set $X=\{x_1,x_2,...x_n\}$ and I want to define a function: $$f(X)=\prod_{j=1}^n{\sum_{i=j}^nx_i \choose x_j}$$ However, in this function I'm treating this set as a sequence, as sets don't ...
0
votes
2answers
36 views

How to interpret $\sum_{1 \le i , j\le n} $?

I have issues with understanding the exact meaning of $\sum_{1 \le i , j\le n} $. For example, how do I have to read the following combination of symbols? $$\sum_{1 \le i , j\le n} a_i b_i \langle ...
1
vote
1answer
22 views

How to solve equations that consist of summations in both sides

I have a function $f$ defined as follows, $$f(n) = \sum_{1 \leq i \leq n} i(i-1)$$ but I want to find $n$ and $n'$ for which the following holds, $$\frac{f(n)}{n(n-1)} = \frac{3f(n'/3)+6f(2n'/3)}{n'...
0
votes
1answer
54 views

Even harmonic sums? [duplicate]

How do we calculate this? $ \displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{2n}}{2n} $ I am stuck that the integrals isn't converging for harmonic (even) numbers . somebody please help .
1
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2answers
59 views

Generating function for $\sum_{n=0}^{\infty} n^{p} x^n$

I am trying to derive the generating function for $H(x,p) = \sum_{n=0}^{\infty} n^p x^n$ I am trying to solve it with the following logic: (Edited now, trying a new framing) Base case: $$H(x,0) = \...
1
vote
1answer
24 views

Need help with inductive proof of Binomial Theorem

I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial. I got stuck trying to read the Induction Proof. They give an example of using the Sum notation: $$ (x + y)^...
1
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0answers
33 views

Integral of a summation related to $\sin$ expansion

I am trying to evaluate the following integral. It has similarity to the Maclaurin expansion for $\sin$. $$\int_{-\infty}^\infty{\sum_{n=0}^{\infty}\frac{(-1)^n}{\left(2n+x^2\right)!}}\text{dx}$$ ...
6
votes
2answers
62 views

Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
0
votes
3answers
150 views

For $\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$, show there is a choice of signs such that it is irrational [on hold]

Considering $$\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$$ where you can replace each $\pm$ with $+$ or $-$. Prove that there is at least one choice of signs such that the number is ...
-2
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1answer
119 views

$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$ change a sign to be rational [on hold]

I have this problem: $$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$$ Prove that you need to change ONLY a sign (to convert a $+$ to $-$) of a single square root, for the sum to be rational. EDIT:...
2
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2answers
55 views

Summing a series of integrals

I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here. I have a series of integrals I would like to sum, but I don't ...
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0answers
14 views

GCD Summation function

I know that GCD summation function ($\sum_{i=1}^{i=n} \gcd(i, n)$ is multiplicative. Thus it can be calculated in $O(\log n)$ complexity using factorization. But I want to want to compute the same ...
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0answers
26 views

Bound on binomial summation

The bound for $\sum_{i=1}^n\binom{n}{i}2^i$ is $O(3^n)$ but what will be the bound for $\sum_{i=1}^{\frac{n}{2}}\binom{n}{i}2^i$. Any idea how should I proceed.
0
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0answers
40 views

Bizarre binomial sum

It is many times that we need to compute discrete convolutions. Driven by this need we have discovered a following formula: \begin{equation} \sum\limits_{l=0}^k \binom{l+A_1}{A_2} \binom{-2 \beta (k-l)...
0
votes
1answer
44 views

Order of summation for shifted exponential function

I want to represent the function: \begin{equation} f(x)=e^{-a(x-b)^{2}} \end{equation} where, $0<a<1$, $x\in\mathbb{R}$, and $b\in\mathbb{R}$. As a power series for an integral I am working ...
2
votes
2answers
39 views

How to prove that a sum of $\cosh(kx)$ is equal to a formula? [duplicate]

I need to prove that $$\sum_{k=0}^{n}\cosh(kx) = \frac{\sinh((n+1/2)x) + \sinh(x/2)}{2\sinh(x/2)}$$ Can you help me out? How do I even start?
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0answers
14 views

Sum of Independent Levy RVs is Levy RV [on hold]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
0
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0answers
31 views

Analytic continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we define a function $f(x)$ by the following sum: $$f(x)= \sum_{n=0}^{\infty} e^{-x E_n}$$ where $E_n$ is a sequence which is at most $O(n)$. It is known $f(x)$ does not form a natural boundry ...
7
votes
2answers
230 views

Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

By trial and error I have found numerically $$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$ how can this result be derived analytically?
0
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0answers
9 views

Evaluating scalar functions of vectors in multidimensional simplices part II

In this question we want to generalize the result from Evaluating scalar functions of vectors in multidimensional simplices. . To be precise we consider a following multivariate sum: \begin{equation} {...
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0answers
51 views

$\sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} =? $ [on hold]

Can anyone sum up this series? $f(z, t) = \sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} . $
5
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0answers
63 views

Closed form for $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n^2+a^2}}$

Do the convergent sum $$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n^2+a^2}}$$ posses a closed form? ($a \in \mathbb{R}$) Special case is known, for $a=0$ one recalls well known alternating harmonic ...
1
vote
1answer
71 views

Closed form for $\sum_{n=0}^{\infty}\frac{x^n}{(n-a)^2+b^2}$

Given $|x|\leq 1$, has the series $$\sum_{n=0}^{\infty}\frac{x^n}{(n-a)^2+b^2}$$ a closed form expression in simple functions? It is known that for $x=1$ from Closed form for $\sum_{n=-\infty}^{\...
3
votes
2answers
80 views

Finite summation including binomial coefficients and double factorials

I came across the following summation: $$ \sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}). $$ $\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$. Mathematica told ...
0
votes
1answer
50 views

Closed formula for ${r \choose 1}+{r \choose 2}\cdots{r \choose w}$ where $w < r$ [on hold]

Let $r,w \in \mathbb{N}$. Are there some formula for the next sum? $${r \choose 1}+{r \choose 2}\cdots{r \choose w}$$ where $w<r$?
0
votes
1answer
15 views

Evaluating scalar functions of vectors in multidimensional simplices.

In question Multivariate sum over a simplex we deal with certain functions of vectors defined in multidimensional simplices. To be specific we are interested in evaluating a following sum: \begin{...
2
votes
1answer
23 views

double summation of conditional variable depending on sum of integer

I am having trouble with taking a certain summation and finding an explicit value for the summation. The summation is: $$ S = \sum_{w=3}^a \lambda_w \sum_{m=w}^a \lambda_m $$ The only information ...
1
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2answers
66 views

Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 - \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by ...
0
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1answer
40 views
+250

Upper-bounding a sum over non-identity permutations

EDIT: Question 1 has been settled (below). The bounty is for question 2. Let $n\geq 3$ and consider the following function $f:S_n\backslash\{e\}\rightarrow \mathbb{R}$ $$f(\sigma)=\sum_{i=1}^n\frac{...
5
votes
2answers
74 views

How to find$\sum_{i,j,k\in \mathbb{Z}}\binom{n}{i+j}\binom{n}{j+k}\binom{n}{i+k}$ for $n \in \mathbb{N}$

Yeah, it's $$\sum_{i,j,k\in \mathbb{Z}}\binom{n}{i+j}\binom{n}{j+k}\binom{n}{i+k}$$ and we are summing over all possible triplets of integers. It appears quite obvious that result is not an infinity. ...
1
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0answers
25 views

Analytic Continuation of Sum $ \sum_{n=0}^{\infty} e^{-b \sqrt n}$

Suppose we have the following function: $$ f(b)=\sum_{n=0}^{\infty} e^{-b \sqrt n}$$ Is there a closed form expression for the analytic continuation of $f(b)$ to $f(-b)$?
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0answers
12 views

Associative notation for Sigma notation

I wonder the following equation does make sense or possible: $(\sum_{k=0}^{L}-\sum_{k=L+1}^{2L})y[k]$ Instead of: $\sum_{k=0}^{L}{y[k]}-\sum_{k=L+1}^{2L}{y[k]}$ In my opinion, advantage of this is ...
5
votes
2answers
37 views

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

How do I see that if $\sum_{n = 1}^\infty \int |f_n| < \infty$, then $\sum_{n = 1}^\infty f(x)$ converges absolutely almost everywhere, is integrable, and its integral is equal to $\sum_{n = 1}^\...
2
votes
1answer
47 views

How to I solve this summation?

I am having trouble solving this summation: $\displaystyle{\quad\sum_{i = 1}^{n}\,\,\sum_{j = 4}^{i} \left(\,\, j + 2i\,\right)}$. I've only gotten this far: $\displaystyle{\quad\sum_{i = 1}^{n}\sum_{...
1
vote
1answer
13 views

Multivariate sum over a simplex

Let $s\ge 0$ and $d \ge 0$ be integers. Let $\left\{ a_\eta \right\}_{\eta=0}^s$ be some positive numbers and let $0 \le \xi_1 \le \xi_2 \le \cdots \le \xi_d \le s$. Finally let $k$ be a strictly ...
4
votes
3answers
44 views

The sum of more than two consecutive natural numbers cannot be prime.

The sum of more than two consecutive natural numbers cannot be prime. Is the statement true and is there any way to prove it? I was able to prove that the sum of an odd amount of consecutive ...
0
votes
0answers
23 views

Logarithm's inequality correctness

It is well known that for , the following holds: Now, given a set of n points, P, is the following term right for every and for every : If so, how can i prove that the term exists? And if it ...
1
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0answers
22 views

Asymptotic Growth of Function of Prime Counting Function

Consider $f(x)$ defined by $$f(x)=\sum_{k=1}^\infty \pi\Big{(}\frac{x}{k}\Big{)}$$ How may one another function $g(x)$ be defined such that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$I have tried $g(x)=c\...
5
votes
1answer
54 views

Difficult looking summation problem

$$\sum_{n=1}^{\infty} \omega(n)(x^n - 2x^{2n} + (-x)^n) = \frac{2x^2}{1-x^4} $$ Where $\omega(n)$ is the number of prime factors of $n$ and $\vert x \vert < 1$
1
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0answers
22 views

Natural Boundary of Sum $\sum_{n=0}^{\infty} e^{-b n^p}$

Is it possible to prove for which $p$ does the sum $$\sum_{n=0}^{\infty} e^{-b n^p}$$ have a natural boundary on the imaginary b axis?
5
votes
0answers
98 views

Summation of $\sum_{n=0}^\infty e^{-\sqrt n}$

Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$