Questions tagged [summation]
Questions using the Sum command, especially for series and other algebraic objects, and related functions such as SumConvergence
1,033 questions
7
votes
2
answers
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Series when differential operator involved
f[n_Integer] := Nest[-D[#, x]/2 &, Cos[x], n]
Sum[f[n], {n, 0, Infinity}](*does not work*)
The sum of that series can be manually found as
$$s(x) = \frac{4}{5}...
- 1,349
4
votes
2
answers
352
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Why does this trigonometric sum show unexpected poles in Plot?
Given the following sum
Sum[Sin[x^2*(2*t - 1)*y]^2/Sin[2*x^2*y*t]^2, {t, 1, L - 1}]
where $x\le L-2$ and $x$, $y$, and $t$ are all positive (non-zero) integers. ...
- 954
2
votes
2
answers
488
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Seemingly simple infinite sum does not converge [closed]
The following summation, when calculated by hand converges to 0:
$$
\sum_{n = - \infty}^{\infty} a^n = 0, n \in \mathbb{Z}
$$
But, Mathematica complains that the sum does not converge:
...
- 605
2
votes
1
answer
220
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Mathematica does not evaluate $\sum_{m=0}^n\binom mi{n-m\choose k-i}={n+1\choose k+1}$
I recently saw q3964942, in which the asker asserted the titled identity
$$\sum_{m=0}^n\binom mi{n-m\choose k-i}={n+1\choose k+1}$$
to which angryavian had commented on 2020-12-28
Shouldn't your ...
- 243
3
votes
2
answers
136
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Reflectedness of PolyGamma depends on sum direction irrespective of Assumptions, requiring pole cancellation for combinatorial sums
I recently wrote the page a little-known permutation generator, in which I investigated sums of products of reciprocals of coordinates over simplex-shaped regions. (These turned out to be the $r$-...
- 243
4
votes
2
answers
274
views
Help with asymptotics on an interesting sum
I recently learned an interesting limit and was trying to understand the asymptotics. Unfortunately I'm getting nowhere.
Take ...
- 43
1
vote
0
answers
159
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Summation contains additional $\frac{\zeta(3)}{8\pi^2}$ term if evaluated without presimplification
As noted in this math.se question from 2018 (which was using WolframAlpha and Mathematica 7.0, though it persists in 14.0.0), Mathematica's evaluation of the following sum depends on where the ...
- 243
1
vote
2
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51
views
How can I include all terms within a sum? [duplicate]
I would like to include terms that factorize a sum within the sum.
For instance, from
k Sum[Subscript[x, i], {i, 1, L}]
getting
...
- 11
1
vote
2
answers
164
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How to output results of the sum, $\sum_{a=1}^2\sum_{b=1}^a\sum_{c=1}^b 2^a3^b 5^c$ into a list (table)?
I'm trying to write a code that can approximate the following formula,
$$\sum_{n=1}^\infty \frac1n=\prod_{i=1}^\infty\frac{1}{1-\frac{1}{p_i}}\tag{A}$$
In this M.SE question I was convinced that $(\...
- 181
1
vote
2
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176
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How to output each summand of $\sum_{n=1}^k \frac1n$ into a list?
From real analysis, I'm trying to learn more about this famous equation in this post of Mathematics.SE:
$$\sum_{n=1}^\infty \frac1n=\prod_{i=1}^\infty\frac{1}{1-\frac{1}{p_i}}\tag{1}$$
I won't ask ...
- 181
2
votes
2
answers
277
views
No method can regularize the sum Sum[k^3*Exp[k], {k, 0, ∞}]
When entering
Sum[k^3*Exp[k], {k, 0, ∞},Regularization->"Dirichlet"]
Mathematica returns the expression unevaluated. But other regularization methods ...
- 3,992
1
vote
0
answers
105
views
How to define a function through a summation of other functions efficiently?
I want to plot the following function:
$F(x, \omega) = \sum_{m = - \infty}^{\infty}\sum_{n = 1}^{\infty} \theta(\omega m - \xi_{mn}) \cos(x \xi_{mn})$
Where $\xi_{mn}$ is defined as the BesselJZero[m,...
- 23
7
votes
2
answers
279
views
Sum of combination of elements of a matrix
Give a $N \times N$ matrix $M$ in MMA, I am interested in finding a very particular sum using its matrix elements:
$ \sum_{i \neq j \neq k \neq l}^{N} M_{ij}M_{jk}M_{kl}M_{li}$,
which is of some use ...
- 1,226
1
vote
1
answer
123
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Efficient nested summation with matrices
Consider a case where I have an array a[i,j]. Each of these a[i,j]consists of a $2 \times 2$ matrix. I would like to compute the ...
- 1,226
0
votes
1
answer
123
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Annoying issue with infinite summation
I want to evaluate a larger number of terms with infinite summations.
For example, there appear summations like
Sum[HarmonicNumber[j]^2/j^4, {j, 1, Infinity}]
...
- 259