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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56 views

Sum of n sums, Permutations of the indices, how to write them in Mathematica?

I was wondering how to write a function $ F (r, q, n, f) $ in Mathematica, defined in this way: $$F(r,q,n,f):=\sum_{i_0=1}^q f(i_0) \Biggl(\sum_{i_1=i_0+1}^{q+1} f(i_1)\biggl(\sum_{i_2=i_1+1}^{q+2} ...
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2answers
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1answer
67 views

Product from max to min [closed]

Product[f[i], {i, 1, 4}] gives us f[1] f[2] f[3] f[4] Is there any way I can take the product it will give something ...
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1answer
32 views

Simplifying sums and showing equality - limitations?

Is it possible to verify the following lhs,rhs involving the sums are equal, with Mathematica? I can verify it for individual values of $d$ variable: ...
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2answers
116 views

How to calculate complicated expression in WolframAlpha

I need to evaluate the limit: $$\lim_{n\to\infty}\prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right).$$ I could not type into WolframAlpha ...
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57 views

Summing over indices

Suppose I have a simple equation with indices, like the one shown in the image. How can I use Mathematica to implement it? I am aware of summing when a specific variable changes value of some range, ...
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0answers
30 views

Simplifying a sum

The expression I'm trying to do something with is Sum[1/(a!(s-a)!)b^(-(2a+1)/2) Sqrt[\[Pi]](2a)!/(4^a (a)!)Pochhammer[-a,k],{a,k,s}] where ...
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1answer
38 views

Mapping function into elements of Sum

I'm trying to set an attribute that's valid for a generic sum, where n and f are arbitrary. de[A_ + B_] := de[A] + de[B] de@Sum[f[i], {i, n}] Is this possible? ...
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51 views

Limit of an infinite summation

The above is from Maple 2019.1. Is there a way to achieve the same result from MMA12? Tried Limit[Sum[Sqrt[1 + k^2/n^3] - 1, {k, 1, n}], {n -> Infinity}] ...
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0answers
34 views

Finding $\frac{4}{l}\sum_{m=0 \\ (j+m-1) even}^{j-1}\frac{3j-2m}{\xi_m}\phi_m(x)$

I have tried this Sum[4 *j* ChebyshevT[m, 2 x/l - 1] /l EvenQ[j + m - 1] , {m, 1,j - 1}, {j, 1, 10}] But I got 0
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3answers
68 views

A question about the use of Sum

Suppose I have the following formula. It calculates the average value of $n$ evenly spaces numbers on the range from $L$ to $H$. ...
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0answers
56 views

Unexpected difference between integral and summation?

I am trying to integrating something like: Integrate[Exp[-i*(k*x+k*z)]*Exp[-(x^2+z^2)],{x,-largenumber,largenumber},{z,-largenumber,largenumber}] My issue is that ...
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1answer
75 views

Expectations of sums

I would like to use Mathematica to derive some bounds on empirical estimators, such as $E[Y]$ where $Y = \tfrac1n\sum_{i=1}^n (X_i - X)^2$ and $X = \tfrac1n\sum_{i=1}^n X_i$. For a moment I thought ...
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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 ...
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1answer
38 views

Speed up summation

I want to evaluate the following sum at lots of different values of $R_0$ and $N$ (roughly 100 of each, R0 ranging from 1 to 6, N ranging from 1000 to 10000): $ \langle Y \rangle = \frac{\sum_{Y=1}^N ...
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0answers
41 views

Discrete Sum of Function over a Region

I might have completely missed something in my search. I want to discretely sum over a function, $f(x_i,y_j)$ multiplied by some other function $g$ over a general region $S$. $$\sum_{(x,y)\in S}f(x,...
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2answers
72 views

How can I help SumConvergence give the right result?

I've been trying to use the SumConvergence on the following series: SumConvergence[1/(n Log[n] Log[n Log[n]]), n] This ...
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2answers
39 views

Problems with double Sum

I have some decent problems with performing a double summation. The Sum is as follows ...
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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 ...
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1answer
41 views

Respecting excluded index in sum

I'm using a function involving a sum where some indices are excluded: ...
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0answers
69 views

Matheamtica Junior on HPC

I am learning Matheamtica on HPC and have never used a linux system before. I have turned the style "input" into "code" and save the file as m format. However, the HPC does not work. The code is ...
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1answer
63 views

Multiply a Sum by a factor

A very simple question: How can I tell to Mathematica that: $\begin{equation} x*\sum_{k=0}^{\infty}\,b_kx^k=\sum_{k=0}^{\infty}\,b_kx^{k+1}\end{equation} $ I tried to multiply but Mathematica gives ...
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1answer
56 views

How does one use NSum within NIntegrate properly?

If I use symbolic integration for the following: Sum[Integrate[i + x, {x, 1, 7}], {i, 1, 7}] 336 as one can see it gives the answer as it seems to '...
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1answer
44 views

Error with NSum : it returns NSum::nsnum: Summand (or its derivative) f[n] is not numerical at point n=17

Consider the following example (I had a lot of trouble to find a minimal working example, I think it is compactified enough now). ...
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2answers
43 views

Sum up different arrays into a new array

I have a question regarding sums in arrays. So I have the following array: ...
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1answer
93 views

Calculation of sum $\begin{aligned}\sum_{k = 1}^{n - 1}\end{aligned}\left(1+\cos\left(\frac{k\,\pi}{n}\right)\right)^n$

Having established that Mathematica cannot calculate the following summation: sum = Sum[(1 + Cos[k Pi/n])^n, {k, 1, n - 1}] I implemented the classic "plan B", ...
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1answer
38 views

Modifying/optimizing a double sum with an If condition

I would like to better understand double summations where one of the sums depends on the upper limit of the previous sum. This appears frequently in representation theory (to the extent of my ...
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2answers
66 views

Apparent contradiction in double summation

I have two expressions which, if my maths is correct, should both be true. But Mathematica doesn't agree. I can take the expression E^(-n^3) out of the single ...
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1answer
59 views

Sum of powers of zero [duplicate]

I would like to calculate the following sum Sum[0^(k-a), {k, 0, Nin}] for a positive integer $a$. With considering $0^0=1$, my expected answer of the sum is $1$, ...
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3answers
100 views

Sum with variable terms to sum over

Suppose I have a polynomial like this: $$a=x_{j_1} + x_{j_1}x_{j_2} + x_{j_1}x_{j_2}x_{j_3} + ...+x_{j_1}x_{j_2}x_{j_3}...x_{j_n}$$ I want to create a function that takes this polynomial and does the ...
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0answers
57 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. ...
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1answer
53 views

Simplify multiple summations involving Kronecker deltas

Sorry if this has been asked before, but I couldn't find a specific answer to it. These work, i.e. they simplify: ...
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2answers
68 views

Collecting terms from expression with indexed functions

Say I have an expansion of terms containing functions y[j,t] and its derivatives, indexed by j with the index beginning at 0 ...
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2answers
151 views

Problem with extracting a constant multiplier out of sum

For a generic symbol A[i] 2 Sum[A[i], {i, 1, n}] == Sum[2 A[i], {i, 1, n}] does not return ...
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1answer
104 views

Checking an interesting result for a sum

If someone is curious I have solved it here: https://math.stackexchange.com/a/3242204/647013 This question is related to this post https://math.stackexchange.com/q/3241994/647013, but I am fairly ...
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24 views

What is the principal difference betwen two results of a seemingly similar sums?

Summing Sum[(a^2 + (b + n)^2)^(-1), {n, -Infinity, Infinity}] gives $$\frac{\pi \sinh (2 \pi a)}{a (\cosh (2 \pi a)-\cos (2 \pi b))}$$ whereas summing <...
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1answer
64 views

What is the meaning of True in my result? [closed]

When I do the sum Sum[(a + (b + π n)^2)^(-1), {n, -∞, ∞}] the result reads $$\begin{array}{cc} \{ & \begin{array}{cc} \frac{\coth \left(\sqrt{a}+i b\...
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0answers
19 views

Incorrect result with very simple in(de)finite sum of conditionals [duplicate]

The results of both Sum[If[OddQ[2 m - 1], 1, 0] q^m, {m, n}] and Sum[If[OddQ[2m-1],1,0]q^m,{m,\[Infinity]}] are 0 (in ...
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1answer
188 views

How to approximate the partial sum formula of a summation to all real numbers using Mathematica?

I am sure one has to use indefinite sums and the Euler-Maclaurin formula. $$\sum_{x=0}^{n} f(x)=\sum_{x}f(n+1)-\sum_{x}f(0)=\int_{0}^{n+1}f(t) \ dt -\frac{1}{2}f(n+1)+\left(\sum_{k=1}^{\infty}\frac{...
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2answers
88 views

Sum not recognized as a linear operator by Solve

When I try to solve an equation with a constant under the summation sign, Mathematica does not factor the constant out of the summation and fails to solve a simple equation. How do I make ...
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1answer
184 views

How to evaluate sum with one million summands?

For a research project I am working on currently, I need to do a very simple and straightforward calculation. Unfortunately, I do not know how to include Mathematica code here, but it is very short ...
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3answers
804 views

How to speed up large double sums in a table?

I am calculating a 3-by-3 matrix whose elements are given as follows: $$ M_{mn} = \frac{1}{N}\sum_{i=1}^N \sum_{j=1}^N (r^i_m - r^j_m)(r^i_n - r^j_n) \tag{1} $$ where $N$ is the total number of ...
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1answer
208 views

Why is the result of a Cesaro regularized sum dependant on a mathematically redundant parameter?

I am trying to determine the series $\qquad \sum_{k=0}^\infty {\rm myCsc}(x,\,\epsilon)\,{\sin(k\,m_C + a_C+\frac\pi 4)}$ where $\qquad myCsc(x,\epsilon))= \begin{cases}i\,\epsilon & 0 \...
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2answers
202 views

How to sum elements of a list of lists?

I know, the title may seem complicated. I have this list: myList = { {1, 0},{2, 3},{4, 1} } I want to sum all the sublists (element by element) to obtain this ...
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1answer
67 views

Using the Sum function square the sums of numbers

How do I use the Sum function for adding and squaring consecutive terms? It's an exercise from Wolfram Challenges. I want to write my own function that uses the Sum function to get the sum of (1+2)^2 ...
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1answer
92 views

Efficiently define a function as the numerical result of infinite sums

I want to approximate the solution to the following sum, in such a way that I can plot the function for the variable $\phi$, for a fixed value of $\mu$. \begin{equation} f(r(\phi), \mu)=4 e^{-\mu ...
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2answers
124 views

FullSimplify a trigonometric expression doesn't work as expected

I know this kind of question is frequent asked, yet each case has its own particularities. I will show my problem. I define the following: ...
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0answers
59 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 ...
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5answers
94 views

Sum Over Solutions to an Equation

Two Related Questions Is there any general built-in functionality for computing a sum over solutions to an equation? This is common in number theory. For example, computing sums of the following form....
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1answer
37 views

Clarification on how Total[] can be used on multi-dimensional array

I have a rank 5 tensor that ultimately I want to modify so that for the first 3 dimensions, each element is summed together. The result will be a rank 2 tensor whose elements are the summed totals ...