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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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11
votes
0answers
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 ...
8
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0answers
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 ...
7
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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[4] Then the ...
5
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0answers
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 ...
5
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0answers
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 ...
4
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0answers
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 ...
4
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0answers
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 ...
4
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0answers
264 views

Does Mathematica know that $\small\frac{\vartheta_3\left(0,\frac{1}{\sqrt[10000000000]{e}}\right)^2}{10000000000}$ 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: <...
4
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0answers
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 ...
4
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0answers
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 ...
4
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1answer
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 ...
3
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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 ...
3
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0answers
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 ...
3
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0answers
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 ...
3
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0answers
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 ...
3
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0answers
104 views

Simplifying Redundant Piecewise Cases

I would like to perform a summation from $1$ to $M$ of a simple piecewise function. For example, ...
3
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0answers
116 views

How to get this terrible summation/product to run in Mathematica?

I've come across this formula and have no idea where to even start. (My assumption here is that $m,n$ are known and input into the expression to arrive at an answer.) $$f(m,n) = \sum_{\substack{0 \...
2
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0answers
44 views

Triple infinite summation of a 3D Fourier series

I'm trying to evaluate the equation below excluding the case when $n_x=n_y=n_z=0$. I know this equation converges everywhere except where x,y, and are all multiples of $2\pi$. I've attempted breaking ...
2
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0answers
58 views

Computing Definite Sums of Rational Functions

I am attempting to compute a rather complicated sum, $S_n$, that in the end satisfies the relation $(S_n + T_n) = (\frac{(n+1)^2 - 1}{n+1})m^3 + O(m^2)$. I should note that $T_n$ is also unknown. ...
2
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0answers
118 views

Why do these sums return the same result when the results should be different?

Sum[1, {k, 1, Infinity}, Regularization -> Dirichlet] gives -1/2, which is right. ...
2
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0answers
59 views

How to evaluate $\sum_{i=1}^{n-k+1} i \binom{n-i}{k-1}$ to get $\binom{n+1}{k+1}$?

I evaluate the following summation using mma (Version: 11.2.0.0): $$\sum_{i=1}^{n-k+1} i \binom{n-i}{k-1}$$ ...
2
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0answers
79 views

Convergent infinite sum fails to converge in Sum[…]

It looks like this. ...
2
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0answers
98 views

How to calculate the sum of two squares of integer-order Bessel function, multiplied by the cosine of the sum of the orders of the Bessel functions?

How to calculate the sum of two squares of integer-order Bessel function, multiplied by the cosine of the sum of the orders of the Bessel functions: \begin{align} \sum _{p=-\infty }^{+\infty } \sum _{...
2
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0answers
103 views

Weird behavior of PolyLog

I have asked Mathematica (11.1.0.0) to calculate Sum[Log[n]/n (-9/10)^n, {n, 2, ∞}] This is the sum $\sum_{n=2}^{\infty}\limits\frac{\log n}{n}\left(-\frac{9}{10}...
2
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0answers
94 views

Error computing sum “Infinite expression 1/0 encountered”

I am trying to calculate a symbolic sum.The expression is defined as follows: ...
2
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0answers
129 views

Why can Mathematica compute numerical sums more efficiently when they are written as matrix operations?

Let $f(n)$ and $K(n,m)$ be functions such that the double sum, which we wish to evaluate numerically, $$ \sum_{n=1}^a \sum_{m=1}^a f(n) f(m) K(n,m) $$ exists when $a$ is some large positive number. I ...
2
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0answers
76 views

Follow-up to “How to differentiate formally?”: Efficiency concern

In link to "how to differentiate formally?" and particularly to the answer by @Jens, I want to do something like this: ...
2
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0answers
388 views

Simplifying symbolic multiple sums

suppose I have a multiple sum with an unspecified number of indexes: $$\sum_{i_1=1}^n \ldots \sum_{i_k=1}^n x_{i_1}\otimes\ldots\otimes \hat{x_{i_j}}\otimes\ldots\otimes x_{i_k}$$ with $x_{i_j}$ ...
2
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0answers
538 views

Computing a sum

I'm trying to make Mathematica compute this sum: Sum[(-1)^k (n - k)^2 Binomial[2 n, k], {k, 0, n}] As is, I get an awful formula: ...
2
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0answers
405 views

Faster ways to compute recursive summation

It takes a long time to compute the summation below, and I'd like to know if there are some better ways to compute things faster. I have used $3$ ways to calculate, but they are very unsatisfactory. I ...
2
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0answers
91 views

Limit[Sum[(2*E*n)^w/(w^(n/2+w)), {w,2,n}],n->Infinity]

I would like to show that the following (and other similar formulae) tends to zero. Limit[Sum[(2*E*n)^w/(w^(n/2+w)), {w,2,n}],n->Infinity] What's the right ...
1
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0answers
60 views

Flaky pattern-matching for Mittag-Leffler sums?

This sum correctly gives the Mittag-Leffler function: Sum[z^k/Gamma[α*k + α], {k, 0, ∞}] MittagLefflerE[α, α, z] Simply factoring the argument of ...
1
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0answers
68 views

Sum over cyclic permutation of indices

To define the Schouten bracket I need to be able to sum over a cyclic permutation of the indices: $$ [\Phi,\Xi]_S=\mathfrak S_{i,j,k} \left(\Phi^{is}\partial_s\Xi^{jk}+ \Xi^{is}\partial_s\Phi^{jk}\...
1
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0answers
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 ...
1
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0answers
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 ...
1
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0answers
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 ...
1
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0answers
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 ...
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0answers
50 views

Symbolic summation with variable bounds and variable number of indices

I wish to compute compute terms like Sum[f[t[j[1]],t[j[2]],...],{j[1],m},{j[2],n},...] for arbitrary positive integer n and any ...
1
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0answers
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 ...
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0answers
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 !!
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0answers
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 ...
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0answers
90 views

How to work out sums symbolically?

Intro Consider these two equal expressions of the variance $$\frac{\sum _{i=1}^n x(i)^2}{n}-\frac{\left(\sum _{i=1}^n x(i)\right){}^2}{n^2}$$ and $$\frac{\sum _{i=1}^n \left(x(i)-\frac{\sum _{i=1}...
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0answers
61 views

ParallelSum faster… if I don't load my own Package

I have a big sum to compute. I remarked that if I load my package, the parallelisation is as slow as the "non" parallelisation. Whereas, when I don't load it, the parallelisation is faster. The ...
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0answers
105 views

How to find analytical summation of Legendre polynomials using Mathematica?

I want to find analytical summation formulas for these 4 types series of Legendre polynomials: ...
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0answers
700 views

Algebra with Sums In Mathematica

I'm trying to perform basic algebra with summations and I haven't been able to find any information on whether it is possible in Mathematica. For instance, I took this rule about multiplying sums ...
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0answers
297 views

Product of modified Bessel functions of first kind and exponential

There is a equation which goes like: $$\sum_{n=-\infty}^{\infty}I_n(x)e^{-x}=1$$ where $I_n(x)$ is the modified Bessel function of the first kind, and $e^{-x}$ is the exponential function. This is a ...
1
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0answers
34 views

Partial Sum of Binary Sequence not Working

I have the following code: ...
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0answers
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 ...
1
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0answers
51 views

Simplify expression?

Suppose I have expressions that involve terms like: p1.p1+p2.p2+p3.p3+p4.p4 p1, p2, p3, and p4 are all elements of an array: ...
1
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0answers
139 views

Problem with calculating Harmonic Numbers

I have tried to take a series of Harmonic Numbers using mathematica but there have been issues in calculations. So far when I computed the sums at a small value range such as ...