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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11
votes
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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 ...
8
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0answers
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 ...
7
votes
0answers
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[4] Then the ...
6
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0answers
64 views

What makes ListPlot better than N?

I wanted to numerically verify the validity of the formula for the first Stieltjes constant $$\gamma_1=-\frac12\sum_{n=0}^\infty\frac1{n+1}\sum_{k=0}^n\binom{n}{k}(-1)^k\log^2(k+1)$$ ...
5
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0answers
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 ...
5
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0answers
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 ...
5
votes
0answers
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 ...
4
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0answers
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 ...
4
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0answers
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 ...
4
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0answers
267 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
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 ...
4
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0answers
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 ...
4
votes
1answer
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 ...
3
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0answers
114 views
+50

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: ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
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0answers
124 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
111 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}...
3
votes
0answers
117 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
30 views

`SumConvergence` gives different results for two equivalent series

Bug persisting through v12.0.0.0 and resolved in v12.1.0.0 Consider these two sequences: a = 2^n/(n! Gamma[(1 - n)/2]^2); b = Simplify[a /. n -> 2 m] It turns ...
2
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0answers
35 views

Factor independent term from a summation?

Can I factor $t^2$ from $\frac{1}{t}\sum_{n=1}^{n1}t^2Exp[-a n^2+bn]$ to get $t\sum_{n=1}^{n1}Exp[-a n^2+bn]$ using some command? ...
2
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0answers
40 views

Summing with assumptions provides unexpected output

I am computing a closed form of a number of sums having generally the following form: $$ \sum_{i = 0}^{m/2}\sum_{\substack{0\leq k \leq i \\ 2k \equiv i + m \, \text{mod } 3}} k $$ It looks like ...
2
votes
0answers
65 views

Why is the sum of the series calculated using the integral

I tried to calculate the sum the Harmonic series via NSum up to a huge limit and got an error: NIntegrate: Numerical integration converging too slowly; suspect one of the following: singularity, ...
2
votes
0answers
47 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
votes
0answers
78 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
votes
0answers
66 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 ...
2
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0answers
123 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
votes
0answers
71 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
88 views

Convergent infinite sum fails to converge in Sum[…]

It looks like this. ...
2
votes
0answers
109 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
102 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
146 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
votes
0answers
394 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
votes
0answers
550 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
votes
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
vote
0answers
21 views

Summing Kronecker Deltas: Sharp slowdowns for simple sums

I'm using Mathematica 12.0 Student Edition. I'm a little confused by the length of time Mathematica takes to evaluate certain sums of Kronecker Deltas or Discrete Deltas. Here's a simple example below:...
1
vote
0answers
58 views

Getting NSum to go to the right depth in recursive definitions

I wanted to produce some plots of the action of the Gauss shift map on cumulative distribution functions. This means I wanted to plot functions $F_n(x)$, for $0 \leq x \leq 1$, defined by $F_1(x) = x$ ...
1
vote
0answers
69 views

Evaluating another symbolic sum

Here I asked about a symbolic sum, and received three very insightful replies (from: მამუკა ჯიბლაძე, Carl Woll and Dr. Wolfgang Hintze) which did the trick. (Thank you again!) Currently I am trying ...
1
vote
0answers
135 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
103 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
vote
0answers
140 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
vote
0answers
273 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
vote
0answers
89 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 ...
1
vote
0answers
54 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
vote
0answers
113 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 ...
1
vote
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 !!
1
vote
0answers
61 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 ...
1
vote
0answers
93 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}...