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

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5
votes
1answer
52 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. ...
1
vote
2answers
48 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) = \...
2
votes
2answers
66 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 ...
1
vote
2answers
352 views

Proving the geometric sum formula by induction

$$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$ I want to prove this by induction. Here's what I have. $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$ I wanted to factor a $q^{...
1
vote
1answer
23 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
vote
0answers
26 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 $\sin$. $$\int_{-\infty}^\infty{\sum_{n=0}^{\infty}\frac{(-1)^n}{\left(2n+x^2\right)!}}\text{dx}$$ ...
-2
votes
3answers
123 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
votes
1answer
110 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
votes
2answers
53 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 ...
-1
votes
0answers
25 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
votes
1answer
41 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 ...
1
vote
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 ...
0
votes
0answers
39 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)...
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?
0
votes
2answers
73 views

Transform double sum $\sum_{i=0}^\infty \sum_{j=0}^i$ to $\sum_{i=0}^\infty \sum_{j=0}^\infty$?

Consider a double sum (assuming it converges) $$\sum_{i=0}^\infty \sum_{j=0}^i f(i,j)$$ Is there a convenient way to rewrite this sum so that both summations go from zero to infinity $\sum_{i=0}^\...
-1
votes
0answers
12 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
votes
0answers
28 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
204 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
votes
1answer
34 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{...
6
votes
4answers
3k views

sum of this series: $\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$

I am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$$ my steps: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=...
0
votes
1answer
14 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{...
0
votes
0answers
5 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} {...
-3
votes
0answers
47 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} . $
1
vote
1answer
69 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}^{\...
5
votes
0answers
58 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 ...
15
votes
4answers
844 views

Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?

If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically ...
1
vote
2answers
64 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 ...
48
votes
16answers
4k views

How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
5
votes
2answers
73 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. ...
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
vote
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 $f(-b)$?
1
vote
1answer
11 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 ...
-1
votes
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
35 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
45 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_{...
5
votes
5answers
118 views

Prove that $\sum_{k=0}^n \binom{3n-k}{2n}=\binom{3n+1}{n}$

Prove that $$\sum_{k=0}^n \binom{3n-k}{2n}=\binom{3n+1}{n}$$ I've tried multiple things that didn't work. Maybe this would help $$\sum_{k=0}^n \binom{3n-k}{2n}=\sum_{k=0}^n \binom{3n-(n-k)}{2n}=\...
0
votes
0answers
22 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 ...
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 ...
1
vote
2answers
108 views

Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion

If $l$ and $n$ are any positive integers, is there a proof of the identity $$\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!\;$$ which uses the Inclusion-Exclusion Principle? (If necessary, ...
11
votes
6answers
777 views

Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. Here's the formula: $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$ Can anyone give a proof of this result? Note:...
1
vote
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
4answers
321 views

Sum of sum of binomial coefficients $\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
0
votes
2answers
55 views

Asymptotics of $ f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k} $

Define $$ f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k} $$ for some fixed constant $c$ (say, $0<c<1/2$). What are the asymptotics of $f_c(n)$ as $n\to\infty$? It seems that this should be ...
0
votes
3answers
107 views

Alternating sum with binomial coefficients $\sum_{k=0}^{49}(-1)^k\binom{99}{2k}$

$$\sum_{k=0}^{49}(-1)^k\binom{99}{2k} = ?$$ I've tried expanding the binomial coefficient in its factorial form and can't seem to get to manipulate it in a way that solves the expression. $C_{99}^{...
6
votes
1answer
306 views

Sum of product of binomial coefficients and exponential function: $\sum^{n}_{k=0}2^k{{n+1}\choose k}{{r-n-2}\choose {n-k}}$

I would like to know how to obtain (if it exists) a closed form expression of the sum $$S=\sum^{n}_{k=0}2^k{{n+1}\choose k}{{r-n-2}\choose {n-k}}$$ So far, I have tried to use the method of ...
0
votes
1answer
153 views

Simplifying sum of binomial coefficients $\sum_{j=m+n+1}^{8S+1}{8S+1 \choose j}$ where $S$ is a half-integer

I'd like to simplify the following sum: $$\sum_{j=m+n+1}^{8S+1}{8S+1 \choose j},$$ where $S\in\{1/2,1,3/2,2,5/2,\ldots\}$ and $\ m,n\in\{1,3,5,7,9,\ldots,4S-1\}$. By simplifying I mean ...
2
votes
0answers
401 views

Double sum with binomial coefficients $\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$

Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$ It's quite likely that it can be done ...
1
vote
2answers
90 views

Finding the summation of a product of the particular binomial coefficients: $\sum_{j=0}^{k} \binom{n-j}{p} \binom{m+j}{q}$

How can I simplify the following expression? $$\sum_{j=0}^{k} \binom{n-j}{p} \binom{m+j}{q}$$ where $n,m,p,q,k$ are positive constants such that $n-k \ge p$ and $m \ge q$.
2
votes
1answer
418 views

A Curious Binomial Coefficient Sum: $\sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -\ell + 1}{n}$

Let $k, \ell \leq n$ be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -\ell + 1}{n} = \binom{n - \ell + 1}{n} \phantom1_{2}\...
47
votes
2answers
4k views

Identity for convolution of central binomial coefficients: $\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...