Questions tagged [summation]
Questions using the Sum command, especially for series and other algebraic objects, and related functions such as SumConvergence
777
questions
0
votes
1answer
20 views
using a list of parameters in NSum
I am trying to make a function that uses NSum, that takes a list of parameters. But NSum doesn't appear to play nicely with lists. A minimum working example is:
...
4
votes
2answers
151 views
How to get the right approximation for a series involving the harmonic number?
The right numerical value of the closed form of $\sum_{n=1}^\infty\frac{4^n H_n^2}{{2n\choose n}n^2}$ is $40.66752074791188333...$.
I tried to verify this result on Mathematica using the command:
<...
1
vote
0answers
73 views
Computation of infinite series containing Zeta function
(this is my first question on this forum I'm totally inexperienced in mathematica)
Consider the given alternating series:
$$f(x) =\sum_{n=0}^\infty \frac{2a_n(x-1)^{2n+1}}{\zeta(-2n-1)}$$
Here, $a_n= (...
0
votes
0answers
21 views
How to assign a summation to existing formula derived from elsewhere? [closed]
I derived an equation algebraically and the solution is in the form of
test1=a*z[t]/(a+x[i]^2+y[i]^2)^(2/7)
To continue to build up the equation, I have to assign ...
0
votes
0answers
33 views
0
votes
1answer
66 views
How would I graph this function on Mathematica?
I could graph 1/n^2, but I don't understand how to graph the fractional part (x).
0
votes
1answer
81 views
Does Mathematica have a problem with sums involving Stirling numbers of the second kind?
In one of my calculations, I run the command:
Sum[(StirlingS2[k - 1, 4] + StirlingS2[k, 4])/6^k, {k, Infinity}]
Surprisingly, ...
0
votes
0answers
42 views
Evaluation of a double summation invovlving hypergeometric and exponential functions
I am trying to compute the following double summation over the indices, $m$ and $n$, which involves the hypergeometric function, ${}_2 F_1$, an exponential function and, factorials as a part of a ...
1
vote
1answer
129 views
How does Mathematica evaluate these sum and integral?
How does Mathematica internally evaluate the following (interrelated) sum and integral, and how does it do the subsequent simplification? (I mean, based on what mathematical facts?)
...
1
vote
1answer
26 views
How do I plot various values of the partial sums for a function with 2 variables?
This is what I have tried so far.
I have a function defined as the below:
...
0
votes
0answers
22 views
Why summation of matrices in the MatrixForm gives wrong result? (an example is given) [duplicate]
I want to calculate the sum of two matrices, doing that in MatrixForm, I get
which is wrong, but writing them as
...
0
votes
2answers
41 views
How to collect coefficients that has summation in ODE
I have the following equation,
equation= x''[t]+a*x'[t]+b*x[t]+c*y[t]+d*u[t]+e*v[t]+Sum[f[i]*x[t]+g[i]*u[t]+h[i],{i,1,n}]+k
When I collect the coefficient of each ...
2
votes
0answers
71 views
How to sum the KroneckerDelta[] in equation?
How to sum the KroneckerDeltas in following equation?
k, p1, ...
0
votes
1answer
59 views
Understanding a “strange” output about a finite sum
Input:
Sum[HarmonicNumber[k]/k^2, {k, 1, m}]
That is
$$\sum_{k = 1}^{m} \frac{H_k}{k^2}$$
Output
I will attach a screenshot for I don't even know how to write ...
2
votes
1answer
40 views
Exponents manipulation
When I type(a^2)^s Mathematica does not give me $a^{2s}$ instead it gives ${(a^2)}^s$. Is there a way to make it print $a^{2s}$.
It made some real difference where ...
2
votes
2answers
108 views
Implementing a formula from a paper
Is this:Table[Exp[-((1250^2) dism11[[n, m]])/Sum[dism11[[i, j]], {i, n, 1250}, {j, m, 1250}]], {n, 1, 1250}, {m,1, 1250}] a correct way to implement:
I am kinda ...
1
vote
2answers
121 views
Exploring Matrix Powers with Wolfram (using Sum Notation)
I am trying to gain an intuition for what algebraically happens to a square matrix (say a $2$-dimensional square matrix) when it is successively multiplied by itself. I have used
...
3
votes
2answers
84 views
Does this sum converge, and why?
Mathematica says the following sum
Sum[(mm Gamma[mm])/
Gamma[-(1/2) + mm] - (mm^(3/2) - (3 Sqrt[mm])/8 - (7 Sqrt[1/mm])/
128), {mm, 1, \[Infinity]}]
...
2
votes
1answer
66 views
How can I calculate the Allan Variance of a list of Data?
I have a list with over 10.000 elements of data. Now I wanna calculate the Allan Variance of this Measurement. The Allan Variance is defined as following:
$$\sigma_y^2(\tau)=\frac1{2\tau^2}\langle(x_{...
0
votes
0answers
48 views
Equations with Tensor product and Ket in Mathematica:
I tried to express this equation in Mathematica:
I defined necessary things:
...
0
votes
2answers
81 views
0
votes
1answer
79 views
speed up symbolic summation
I have the following summation
L=24;
sind=Range[-Pi,Pi,2*Pi/L];
Sum[f, {x, sind}, {y, sind}, {x1, sind}, {y1, sind}, {x2, sind}, {y2, sind}]
where ...
0
votes
1answer
28 views
Expand product of symbolic sums (with identically named indices)
I have an expression that is a product of two symbolic sums.
Sum[a[k], {k, 1, n}]*Sum[b[k], {k, 1, n}]
How can I expand this expression? I want to see something ...
3
votes
4answers
222 views
Trouble with the numerical evaluation of a series
The series is $\;S=\displaystyle{\sum_{n=0}^\infty 2^{-n+\sin(n\pi/5)}}$.
Mathematica doesn't find a closed form for Sum[2^(-n + Sin[n Pi/5]), {n, 0, Infinity}], so ...
3
votes
2answers
237 views
How to calculate this summation
Ī· = 0.05211184484645051`;
Sum[Ī·/(r + 1) + ArcTan[Ī·/(r + 1)]
$r$ start from $0$.
How to calculate this summation until it becomes less than 10^-5?
6
votes
0answers
103 views
Sum cause a Recursion problem
Sum[((-1)^(i + 1)*Binomial[n, i]*(n - i)!)/n!, {i, 1, n}]
This cause a Recursion problem.
As the comment said, If simplify the formula and then sum it, problem ...
0
votes
1answer
59 views
simplifying indefinite sums containing Kronecker Deltas
I want to simplify indefinite sums containing KroneckerDeltas, e.g:
$\sum_{k,k1,q} \beta(q) \beta(k+k1+q) \delta(k1+q)= \sum_{k,q} \beta(k)\beta(q)$ where $k,k1,q \;\epsilon \; \mathrm{R}$
...
0
votes
0answers
29 views
summation involving Kronecker Deltas
I see that:
Sum[u[q] u[k] v[k1] v[k + k1 + q] KroneckerDelta[k1 - q], {k1, -\[Infinity], \[Infinity]}]
produces the correct result: ...
2
votes
1answer
61 views
Numerical sum optimization
I need to compute the following sum as fast as possible:
$$P_{ijkl}=\sum_{p,q,r,s}^nW_{sqpr}H_{js}G_{rl}A_{ipkq}$$
I came up with this code:
...
8
votes
2answers
370 views
Cannot reproduce well-known summation result
Bug introduced in 4.2 or 5.0, persisting through 12.2.0.
Note: this is a repost of my OP on math.SE. I am posting it here because multiple users with different Mathematica versions have given me the ...
2
votes
4answers
65 views
Periodical sum of rows by a certain step
I've been struggling with one problem. I have a classic matrix in this example 12x4 such as:
matrix = Table[i, {i, 12}, {4}]
...
3
votes
2answers
76 views
Vectorization of multifold summation to speedup
I searched this website but didn't find any suitable answer describing how one can speed up summation in Mathematica using vectorization techniques and other techniques.
I often have to numerically ...
-1
votes
1answer
61 views
Why can't Mathematica confirm this simple identity? [closed]
It is well known that for positive integer $n\geq2$,
$$
\sum_{k=1}^{n-1} \frac{(-1)^{k-1}(k-1)!^2}{(n^2-1^2)\ldots(n^2-k^2)}=\frac{1}{n^2}-\frac{2(-1)^{n-1}}{n^2 \binom{2n}{n}}.
$$
This identity ...
3
votes
1answer
111 views
Infinite summation giving weird results
We are searching in our group for closed forms of derivatives of hypergeometric functions. This leads to expressions like
$\sum\limits_{m=2}^\infty \frac{z^m\Gamma[m-1/2]H_m}{2m^2\sqrt{\pi}\Gamma[m]}$
...
4
votes
2answers
367 views
Faster, More Elegant Way to Produce a Recursive Sequence of Rational Numbers [closed]
I am studying a recursion below:
$$B_{N,0}=1$$
$$B_{N,k}=-\binom{N+k}{k}^{-1}\sum_{j=0}^{k-1}\binom{N+k}{j}B_{N,j}$$
Now I'm not great at writing in Mathematica. It's been a while since I've used it. ...
0
votes
1answer
86 views
$\sum_{n=1}^{+\infty}\frac{\cos(\frac{n\pi}{2})}{n}$ gives wrong answer in Mathematica? [closed]
I wanted to compute $\sum_{n=1}^{+\infty}\frac{\cos(\frac{n\pi}{2})}{n}$ on Mathematica and it says the sum does not converge, even though it does by Dirichlet's test. When I plugged the same sum in ...
0
votes
1answer
63 views
Analytical expression for a limit
I have the following equation
$$\Delta v = -g I_\mathrm{SP} \left( 1 + \frac{1}{24} \frac{\mu}{r^3} \frac{g^2 {I_\mathrm{SP}}^2}{T^2} \frac{{m_p}^2}{n^2} \right) ^{-1} \sum _{k=1} ^n \log \left( \...
2
votes
0answers
56 views
Evaluate fractional derivative
I needed to evaluate the following fractional derivative of $f(t) = e^{t^2}$. The fractional derivative that I'm currently studying is the Grunwald-Letnikov fractional derivative, which is defined as ...
2
votes
4answers
190 views
Usage of 'Less than' condition in Sum
I have following sum: $$ \sum_{\substack{n,j\\j<n}}^{3} x_nx_j$$. How can I give this $j<n$ condition in Sum?
0
votes
0answers
30 views
Calculating a sum that's taken over all of the k-element subspaces of an n-element space
Currently I constructed a formula (the idea from which it arose was completely combinatorial, won't bother you with that) and was wondering whether this is applic-able in Mathematica (I've used it so ...
0
votes
2answers
35 views
Plotting partial sum with complex number
$$\sum_{n=1}^{N} e^{2\pi in\sqrt2}$$
We have to use Accumulate and ListLinePlot, and my implementation so far with upper bound ...
0
votes
1answer
82 views
Difficulty plotting sum of functions
I'm interested in plotting this function (provided in equation 11 of this paper):
$$
\begin{aligned}
\begin{aligned}
S_{S}(\omega)=& \sum_{m_{5}, m_{I}, m_{S}^{\prime}, m_{I}^{\prime}=-\infty}^{\...
3
votes
4answers
392 views
How to find the sum of that series related to Legendre functions of the second kind?
I mean $$\sum _{n=0}^{\infty } \frac{Q_n\left(\frac{\sqrt{2}}{2}\right)}{n+1}. $$
It's unclear to me whether the series under consideration converges. I have strong doubts concerning its closed form. ...
0
votes
2answers
56 views
How to make calculations involving symbolic summation?
I want to minimize a function involving the sum of $n$ fixed (but indetermined) $x$ values, for example:
Minimize[Sum[(x[i] - a)^2, {i, 1, n}], a]
Expected answer: ...
3
votes
0answers
62 views
Algorithm used by Mathematica for evaluating partial sums
Today, while using mathematica, I entered the command Sum[1/Factorial[n], {n, 0, x}] and found that:
$$\sum_{x\geq n\geq0}\frac{1}{n!}=\frac{e\Gamma(x+1,1)}{\Gamma(...
1
vote
0answers
35 views
Sum over certain indices not working
Define a function $h$ by
h[m_,n_]:=2^n Binomial[m,Ceiling[m/2]]Binomial[n,Ceiling[n/2]]Ceiling[m/2]Ceiling[n/2](m-1)b^(m+n)/((m+n)m!n!)
I am trying to evaluate
<...
4
votes
3answers
128 views
The symbolic result does not give a proper answer when inputs are specified
Define F by
F[m_, k_, i_, j_] := (-1)^(m+k)/(m!*k!)*2^m*Binomial[m, i]*Binomial[k+1,j]
I am trying to find this sum:
...
0
votes
1answer
23 views
Binomial sum only expanded, not computed [closed]
I want to compute this sum: $\sum _{n=1}^4 \frac{(2 n-1) \sum _{k=0}^{n-1} \text{Binom}(6,k) \text{Binom}(10,-k+2 n-1)}{\text{Binom}(16,2 n-1)}$
...
1
vote
2answers
71 views
To write mathematica code for summation using a given data
I am given a set like some terms from the partition of $20$:
$\{\{17, 3\}, \{13, 7\}, \{11, 3, 3, 3\}, \{7, 7, 3, 3\}, \{7, 5, 5, 3\}, \{5, 5,
5, 5\}\}$.
From the above data,
$20= \underline{1}*17+\...
0
votes
2answers
53 views
Computing the coefficients of series
I am trying to fully expand the following
$$\sum_{n=0}^{\infty} \frac{q^{\frac{n(n+1)}{2}}}{(q;q)_n}$$
Which when expressed in Mathematic is: q^(n*(n + 1)/2)/QPochhammer[q, q, n]
I would like to be ...
