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Questions tagged [summation]

Questions using the Sum command, especially for series and other algebraic objects, and related functions such as SumConvergence

92 questions with no upvoted or accepted answers
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97 views

What does that output of Sum mean?

I made the computation ClearAll["Global*"]; r = Sum[1/2^(k*n/(k + n)), {k, 1, 2*n}, Assumptions -> n ∈ Integers && n > 0] and got ...
244 views

Incorrect evaluation for Thue-Morse signed harmonic series

I would like to evaluate $$s = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}+\frac{1}{7}-\frac{1}{8} - ... + \frac{(-1)^{\textrm{binary digit sum}(n-1)}}{n} + ...$$ where ...
117 views

Converting a sum into Σ notation in output

How can I convert the sum into Σ notation in output? For example, I have an array of Array[k,4]. My input is Array[k,4] Sum[K[i], {i, 1, 4}]-K Then the ...
60 views

TransformedDistribution using $k$ iid random variables, but $k$ not fixed

Can I create a TransformedDistribution that uses $k$ independent identically distributed (i.i.d.) random variables where $k$ is not fixed? This question is closely ...
75 views

SumConvergence fails in version 10

SumConvergence[(-1)^(n + 1) ((Cos[n^2] + Sin[n + 2])/7^n), n] Mathematica fails to provide a result (true/false) but wolfram alpha works. What should I do ? It ...
63 views

Double summation giving unexpected result

The expression (in a notebook with Wolfram Mathematica 12.0.0) Sum[s[i, j] - s[j, i], {j, b}, {i, b}] Produces the result 1/2 b EulerPhi[b] Can anyone ...
287 views

Simplify hypergeometric function

Theorem: Let $H_n$ be the $nth$ harmonic number. Then $$\sum_{n=1}^\infty \binom{2n}{n} H_n x^n=\frac{2}{\sqrt{1-4x}}\log\bigg(\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}} \bigg)$$ How can I simplify ...
264 views

Does Mathematica know that $\small\frac{\vartheta_3\left(0,\frac{1}{\sqrt{e}}\right)^2}{10000000000}$ not equal $\pi$

The following is not an identity but is correct to over 42 billion digits: $$\bigg(\frac{1}{10^5}\sum_{n=-\infty}^{\infty}e^{-\frac{n^2}{10^{10}}}\bigg)^2=\pi$$ I want to check this. I tried: <...
225 views

Can a single Sum with multiple iterators be different from nested Sums?

Multiple sums are documented with two or more iterators, for example: Sum[1/(j^2 (i + 1)^2), {i, 1, Infinity}, {j, 1, i}] however the same answer can be ...
185 views

Multiple Constrained Sum

I need to perform a multiple summation, that obeys some conditions. This arises in the study of a statistical physics model. q=Exp[-β]. I would like to ask if ...
114 views

How to make sum of divergent series to use any regularization that succeeds?

For instance, Sum[x, {x, 1, Infinity}, Regularization -> Dirichlet] Sum[Exp[x], {x, 1, Infinity}, Regularization -> Borel] works, but ...
34 views

Why does this sum return unevaluated when testing for convergence?

Why does Mathematica return unevaluated for SumConvergence[Sin[\[Pi]/n^2], n]? I tried looking at the documentation, but I can't find any possible issues. The ...
171 views

On Ж, and the fine-structure constant

I'm trying to reproduce the results from a certain infamous paper that has been moving around the web for the last few days. The details are irrelevant. This paper claims to have a closed-form ...
126 views

How to find the closed form of a relatively simple sum?

I'm trying to find the sum $\sum_{n=1,3,5}^\infty \frac{8}{n \sinh (n \pi)}$ Sum[8/(Sinh[n*Pi]*n), {n, 1, Infinity, 2}] which I know is ln(2), but Mathematica ...
69 views

Wrong computation of a series with FactorialPower`?

I wanted to compute the series defined by $$\sum_{k=1}^\infty\frac{(-1)^{k+1}}k x^\underline k$$ where $x^\underline k:=\prod_{j=0}^{k-1}(x-j)$ is a falling factorial. Thus I write ...
104 views

Simplifying Redundant Piecewise Cases

I would like to perform a summation from $1$ to $M$ of a simple piecewise function. For example, ...
116 views

89 views

Finding general forms of sums in Mathematica

A general form can be obtained for some sums with binomial coefficients. Mathematica is able to find these general forms for some of the simplest sums. For others the output contains for instance ...
96 views

How to sum over the variable in partial derivative operator?

I need to use the partial derivative operator in Wolfram Mathematica within a summation, specifically to define the D'Alembertian operator of scalar fields. I am having trouble summing over the D ...
165 views

Sum over multiindex

I would like to calculate a sum over some multi-indices, that follow a specific pattern. $$\sum\limits_{1\le i_1<i_2<...<i_k\le N} A(k,i_1, i_2, ..., i_k).$$ $A$ is a fix expression of the ...
72 views

Infinite sum with recursive coefficients

Mathematica can handle infinite sums like Sum[ x^k/k!, {k, 0, Infinity}] (* Exp[x] *) Suppose I only know a recursive definition of the coefficients ...
50 views

Symbolic summation with variable bounds and variable number of indices

I wish to compute compute terms like Sum[f[t[j],t[j],...],{j,m},{j,n},...] for arbitrary positive integer n and any ...
100 views

Sum involving hypergeometric function $\mbox{}_1 F_2$

Trying to simplify the sum $$\sum\limits_{n=0}^{\infty}\dfrac{z_1^n}{n!} {}_{1}F_{2}(1;a+n,1-a+n;z_2),$$ where $a\in(0,1)$, $z_1,z_2>0$, and ${}_{1}F_{2}$ denotes the appropriate version of the ...
54 views

Summation with skipping a term

how do i sum the following in Mathematica: the m-th term is 1/(m^2 - n^2), the sum is over odd m and m <> n. i know the answer is -1/(4*n^2) or something like that. thanks !!
60 views

Why MMA cannot get the closed-form of this infinite series of B when it known that of A+B and A?

The following input s1 = Sum[(2 l + 1)*t^l*LegendreP[l, Cos[θ]], {l, 0, Infinity}, Assumptions -> {t < 1, 0 < θ < π}] returns the closed-form ...
90 views

34 views

Partial Sum of Binary Sequence not Working

I have the following code: ...
356 views

Assumptions aren't working in this sum

I have a sum that should be real: FullSimplify@Sum[1/(α n^4 + β), {n, 1, ∞}, Assumptions -> α > 0 && β > 0] But the result have involved even ...