# Questions tagged [summation]

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

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### 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 ...
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### Sum causes a recursion problem

Bug introduced in 8 or earlier and Fixed 13.3.1 Sum[((-1)^(i + 1)*Binomial[n, i]*(n - i)!)/n!, {i, 1, n}] This cause a Recursion problem. As the comment said, If ...
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### 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 ...
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### 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 ...
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### What makes ListPlot better than N?

I wanted to numerically verify the validity of the formula for the first Stieltjes constant $$\gamma_1=-\frac12\sum_{n=0}^\infty\frac1{n+1}\sum_{k=0}^n\binom{n}{k}(-1)^k\log^2(k+1)$$ ...
• 3,104
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### 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 ...
• 181
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### 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 ...
• 155
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### How to automate calculations of sums with harmonic numbers

I would like to ask if anyone knows how can we automate calculations of the following sum Sum[HarmonicNumber[n]^2/((n + 1) 2^n), {n, 1, ∞}] The analytic result is ...
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### 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 ...
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### 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 ...
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### 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: <...
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### 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 ...
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### 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 ...
255 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 ...
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### Weird summation result computing the expectation of Poisson random variable

Can you explain the obtained difference between these two rapidly convergent infinite sums? The correct answer is $-1/2$, and this should be obtained in both presented cases. ...
151 views

### Summation question?

I have following summation that I want to implement using Mathematica: ...
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### Summation variables aren't recognised as dummy variables

I'm trying to write this expression in Mathematica, and calculate the following quantity However, when I tried the following ...
• 241
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### Analytically express chi-square sum and solve

I have an expression: chisq[f_, x_, y_, e_, pars__] := Sum[((y[k] - f@@Join[{x[k]},pars])/e[k])^2, {k,1,n}] which should work for any function that has ...
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### Sum doesn't work as I expected

I'm trying to sum a bunch of polynomials. I'm implementing that: Sum[Subscript[P, k][c] (k + x) z^(k + x), {k, -x, -1}] and mathematica gives me back that I ...
• 165
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### Take symbolic derivative of summation expression and display as unexpanded sum

My main problem here is two-fold: Take symbolic derivative of summation expression Stop summations from being "unrolled"/expanded, e.g., displayed as the sum of terms I also have some ...
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### Can we approximate a matrix power series like NSum does?

Essentially, the following does not work, and I'm wondering if it can be made to: NSum[ MatrixPower[B,n], {n,0,∞}] (Here B is a ...
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### NSum complaining of being non-numerical

I am trying to evaluate the sum $\sum_{k,l=1}^{30}\frac{1}{(k^2+l^2+1)^{5/4}}$ so I write NSum[1/(1 + k^2 + l^2)^(5/4), {k, 1, 30}, {l, 1, 30}] but I get a message ...
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### How to intelligently use FullSimplify and FunctionExpand to simplify complex sums

I am trying to find a compact form of some sums which is related with some Bayesian probability factor (not so relevant, if required further explanation please ask). The point is that I know that the ...
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### How to sum the KroneckerDelta[] in equation?

How to sum the KroneckerDeltas in following equation? k, p1, ...
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### Evaluate fractional derivative

I needed to evaluate the following fractional derivative of $f(t) = e^{t^2}$. The fractional derivative that I'm currently studying is the Grunwald-Letnikov fractional derivative, which is defined as ...
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### Which Method is used during the processing of Sum Function?

I am trying to find out which Method is used while evaluating the following function: Sum[1/((n^2)), {n, 1, Infinity}] Any help would be greatly appreciated. A ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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:...
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### 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 ...
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### 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. ...
751 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 ...
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### 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. ...
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### 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): $$\sum_{i=1}^{n-k+1} i \binom{n-i}{k-1}$$ ...
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### 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 _{...
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### 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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### Error computing sum "Infinite expression 1/0 encountered"

I am trying to calculate a symbolic sum.The expression is defined as follows: ...
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### 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 ...