# Questions tagged [summation]

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

104 questions with no upvoted or accepted answers
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104 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 ...
247 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 ...
121 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 ...
64 views

### What makes ListPlot better than N?

I wanted to numerically verify the validity of the formula for the first Stieltjes constant $$\gamma_1=-\frac12\sum_{n=0}^\infty\frac1{n+1}\sum_{k=0}^n\binom{n}{k}(-1)^k\log^2(k+1)$$ ...
68 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 ...
65 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 ...
78 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 ...
135 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 ...
340 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 ...
267 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: <...
250 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 ...
212 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 ...
141 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 ...
114 views
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### Discrepancy between the results of NIntegrate with different methods and options

I am trying to perform a numerical integration on a function defined through a sum of exponential terms. The summation is given by: ...
37 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 ...
182 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 ...
75 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 ...
I would like to perform a summation from $1$ to $M$ of a simple piecewise function. For example, ...