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

learn more… | top users | synonyms

12
votes
0answers
615 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
10
votes
0answers
238 views

Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
9
votes
0answers
282 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
7
votes
0answers
300 views

How find this sum $\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$

Question: Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$$ where $$L_{n}(k)=1-\dfrac{1}{2^k}+\dfrac{1}{3^k}-\cdots+\dfrac{(-1)^{n-1}}{n^k}$$ since ...
6
votes
0answers
71 views

Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
6
votes
0answers
95 views

Why use Einstein Summation Notation?

Einstein summation convention dictates that repeated indices should be summed. Thus the equation $a_{ij} = b_{ik}c_{kj}$ is taken to mean $a_{ij} = \sum_k b_{ik}c_{kj}$ where in both cases the range ...
6
votes
0answers
99 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot ...
6
votes
0answers
157 views

How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$ \sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
6
votes
0answers
154 views

Second derivative of Hypergeometric function

I'm looking for the following second derivative $$ \kappa_2 := \left . \frac{d^2}{d\lambda^2} \ln \left({_2F_1}\!\left(\tfrac{1}{2},\,- \lambda;\,1;\,\alpha\right)\right) \right \vert_{\lambda = ...
6
votes
0answers
68 views

Series with $k^2$ coefficients

Let $\{S_k\}$ be a sequence of numbers. Then by reversing the sum we have the following, $$A=\sum_{k=0}^n k\cdot S_k \cdot S_{n-k} = \sum_{k=0}^n (n-k) \cdot S_{n-k} \cdot S_{k} .$$ Thus adding these ...
5
votes
0answers
58 views

Heuristics of the sum of squared naturals $(1^2 + 2^2 + 3^2 \cdots + n^2)$

I'm new and this is my first question (though I've been lurking). English is not my native language. Studying on my own. I'm really interested in deriving the formula $1^{2} + 2^{2} + 3^{2} + \cdots+ ...
5
votes
0answers
69 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
5
votes
0answers
62 views

A summation involving the ceiling function

I'm trying to find a better method of calculating the sum $$\sum_{k=1}^N\lceil ak\rceil^2$$ where $a$ is an irrational number. So far, my only idea is to somehow use a best rational approximation. ...
5
votes
0answers
72 views

How to simplify $\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$

Let $$x=\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$$ where $n,k\in\mathbb{Z}^+$. How to simplify $x$? I simplified it for $k=1,2,3$ and I got $n$, $\dfrac12n(n+1)$ ...
5
votes
0answers
71 views

Subset Sum Problem Variation?

There are $100$ cards with a unique number from $1$ to $100$ written over them. How many ways can someone pick exactly $5$ cards where the numbers on them sum to $100$? I am not sure but this could ...
5
votes
0answers
500 views

Proof for an identity involving a sum of binomial coefficients

I am moving through a On The Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my skills are ...
5
votes
0answers
96 views

Sum over squared index

Is there any way to compute the finite series $$S_M = \sum_{n=1}^{M} r^{n^2}, $$ for some real $r$, integer $M$?
4
votes
0answers
42 views

sum of center binomial coefficients over exponential

I'm trying to find a closed form for the following sum, if anyone knows a way, a hint would be much appreciated... $$ X(n) = \sum_{i=1}^n \frac{i \choose \left \lfloor{i/2}\right \rfloor }{2^i}\ $$ ...
4
votes
0answers
72 views

Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
4
votes
0answers
65 views

If $f$ is integrable, then $\sum\limits_{n\ge 1}\frac{1}{\sqrt n}\vert f(x-\sqrt n)\vert$ is almost everywhere finite

I would like to show that $$\sum_{n\ge0}\left\vert \frac{1}{\sqrt n} f \left(x-\sqrt n \right)\right\vert \tag{$*$}$$ converges for almost every (a.e.) $x$. The only technique I have is based on the ...
4
votes
0answers
151 views

Proving an equation involving binomial coefficients

Prove that $$\sum_{q=0}^v \binom{v}{q}\frac{q!}{v^{q+1}} = \sum_{q=0}^{v-1} \binom{v-1}{q} \frac{(q+2)!}{v^{q+2}}$$ Thanks. Below are what I have tried: Approach 1: $$\sum_{q=0}^{v-1} ...
4
votes
0answers
230 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
4
votes
0answers
96 views

Simplifying a sum?

Define polynomials $P_{j,s}^{(r)}$ via the generating series $$\left(\frac{d^s}{dz^s}f(z)\right)^r=\sum_{j=0}^{\infty} P_{j,s}^{(r)}z^j,$$ where $r\geq 1$. Here, $f(z)=z+a_2z^2+a_3z^3+\cdots.$ I was ...
4
votes
0answers
159 views

Difficult Sum: Any Tips?

I'm trying to integrate a very difficult expression, and I've arrived at a particular step involving this sum, which I want to turn into a closed expression so that I can raise it to a power, re-sum ...
4
votes
0answers
132 views

Binomial-like sum involving falling factorials

We know that $\sum_{k=0}^n a^k \frac{n^{\underline k}}{k!} = (1+a)^n$. Is there a known (preferably closed) form for $\sum_{k=0}^n a^k n^{\underline k}$?
4
votes
0answers
82 views

Sums over the positive integers

Can anyone see why it is that if $a$ is large, then $$\log (\sum_m\sum_n \exp(-knm/a)))$$ where $k$ is a constant and $n,m$ take values $1,2,3,...$, can be approximated by $${a\pi^2\over 6k }$$? ...
4
votes
0answers
203 views

Summation of the series $s_b(p)=\sum_k b^{k^p}$ by a double sum in a sense like Ramanujan-method

From some older context I am re-considering the following variant of the geometric series $$ s_b(p)=\sum_{k=1}^\infty b^{k^p} $$ for the convergent cases $0 \lt b \lt 1$ and $ 0 \lt p$ first. I'm ...
3
votes
0answers
33 views

Partial sum of squared binomial coefficients

Is there any formula for the partial sum of squared binomial coefficients $$S_n(k):= \sum_{i=0}^{k} \binom{n}{i}^2,$$ where $k<n$? Thank you in advance.
3
votes
0answers
44 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
3
votes
0answers
54 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
3
votes
0answers
40 views

Is it possible to eliminate the inner sum to evaluate numerically?

Any hints on how to simplify the following double sum to be able to find the sum at least numerically? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$ ...
3
votes
0answers
142 views

Sum problem involving Factorials

got the following problem to prove for $n \in \mathbb{N}$ and $1 \leq i \leq n$: \begin{equation} \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! ...
3
votes
0answers
69 views

Asymptotics of integer compositions

A (weak) composition of a positive integer $n$ into $k$ parts is an ordered sequence of nonnegative integers $(a_1, a_2, \ldots, a_k)$ such that $ \sum_{i=1}^k a_i = n $. I am interested in the case ...
3
votes
0answers
42 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
3
votes
0answers
49 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
3
votes
0answers
72 views

If $A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is,

If $\displaystyle A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\cdots\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is, where $\lfloor A\rfloor = A-\{A\}.$ ...
3
votes
0answers
63 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = ...
3
votes
0answers
26 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...
3
votes
0answers
56 views

Random Wolfram|Alpha identity $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
3
votes
0answers
106 views

conjecture about prime numbers and distance between them

is there a name for this conjecture? Conjecture: given $p_n$ a prime number sequence where $p_1=2,p_2=3,\cdots$, then for all $n\in\mathbb{N}^*$ and $k\in\mathbb{N}$, holds that $\displaystyle ...
3
votes
0answers
130 views

Sum of product of binomial coefficients and exponential function

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 ...
3
votes
0answers
190 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
3
votes
0answers
83 views

Evaluation of the double sum

Is there a way to get a closed expression for the double sum: $$\sum\limits_{n = 1}^\infty \sum\limits_{m = 1}^\infty \left( \frac{m}{(n^2 + m^2 - E_1)(n^2 + m^2 - E_2)} \right)^2$$ , where $E_1$ and ...
3
votes
0answers
146 views

Proving $\sum_{k=1}^{n}\binom{n-1}{k-1}{\binom{n+k}{k}}^{-1}=\frac 12$ combinatorially

Question : How can we prove the following equations combinatorially? $$\begin{eqnarray}\sum_{k=1}^{n}\frac{\binom{n-1}{k-1}}{\binom{n+k}{k}}&=&\frac ...
3
votes
0answers
77 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq ...
3
votes
0answers
66 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
3
votes
0answers
44 views

How to show that $\sum\limits_{k=0}^{\lfloor0.999n\rfloor}\binom{2n}{k} < \binom{2n}{n} $ holds for large n

It seems logical to me since $\binom{2n}{n}$ is in the middle of the row in pascal triangle; therefore, the largest, and for large n the sum adds only the small ones on the left. But I do not have any ...
3
votes
0answers
86 views

Find generating function For sequences

Can anyone out here help? The exercise says: "Find the generating function for each of the sequences below (the general term is given)" Now, the question is how do you find one for those: a) $U_n = ...
3
votes
0answers
104 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
3
votes
0answers
352 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...