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

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21
votes
4answers
2k views

how to differentiate formally?

I have been wrapping my head around this for a while now and I have not found a solution so far. I want to work with an arbitrary number of variables in mathematica and use some built in functions. ...
17
votes
2answers
2k views

Sum or Product with Exclusions

Is there a built-in feature for handling things like: $$\sum_{\substack{i=0\\i\ne j}}^n\frac{a-a_i}{a_i-a_j}$$ and $$\prod_{\substack{i=0\\i\ne j}}^n\frac{a-a_i}{a_i-a_j}$$ or should I work out some ...
13
votes
1answer
249 views

Baffling increase in runtime

Background of my question I discovered Project Euler today, and decided I would work through the problems in Mathematica. I became obsessed with the first problem, which is essentially "sum all the ...
12
votes
5answers
716 views

Double series over primes

I'm very curious if the following double series over primes has a closed form: $$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$ where $\mathcal{P}$ denotes the set of all ...
11
votes
3answers
377 views

Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
11
votes
2answers
190 views

Fastest way to sum the upper triangle

I feel like this is an recurring question: if there's a symmetric matrix whose diagonal is not all 0, how could I get the sum of the part of it that's above the diagonal as fast as possible? Small ...
11
votes
2answers
254 views

How to implement symbolic Ramanujan's summation in Mathematica?

How to implement Ramanujan's summation in symbolic form in Mathematica? For instance, I want as input the function $f(x)=x$, as output $-1/12$, as input $f(x)=1/x$, as output $\gamma$ (Euler's ...
10
votes
2answers
3k views

How do you put conditions on indices in a sum?

I'm relatively inexperienced with mathematica, so I apologize if this is a trivial question. I want to take a double sum over a function $f(i,j)$ of two indices, of the form $$ \sum_{i = ...
10
votes
2answers
249 views

Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$

I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$ I tried ...
9
votes
4answers
1k views

Ways to compute inner products of tensors

One way to evaluate the following sums is combining Table and Sum: $u_{abcd} = \sum_{e=1}^3 v_{aeb}w_{ced}$ $q_{ab} = \sum_{d,e=1}^3 v_{d e a}w_{deb}$ It will look like ...
9
votes
6answers
427 views

Performing Computations on Sets

I would like to find a permutation of $\quad S=\{\frac{1}{10}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{2}{3} \}\quad$ that maximizes the sum of theses elements raised to unique powers: ...
9
votes
2answers
193 views

sudden increase in timing when summing over 250 entries

Using math 9.0.1 on a MacBook Air, I see a sudden increase of Timing by a factor of thousands when I sum over 250 elements of a matrix rather than over 249. So for ...
9
votes
1answer
121 views

Bug in GeneratingFunction?

Bug introduced in 7.0 and fixed in 9.0.0 According to the documentation GeneratingFunction[a[n],n,x]==Sum[a[n]x^n,{n,0,Infinity}] However, for $a_n=1/(n+2)$ I ...
8
votes
6answers
717 views

Alternating sum

A frog is at the bottom of a 30 metre well. Each day it climbs 5 metres up the side, but it then slips back 3 metres each night. How long does it take to reach the top of the well? Is there an easier ...
8
votes
5answers
2k views

Sum all numbers from 1 to 1000 divided by either 2,3,5 or 7

How do I find the sum all numbers from 1 to 1000 divided by atleast one of 2,3,5 or 7? EDIT: I am sorry for complicating this, but I need it to work for 10^11. So anything that requires too much heap ...
8
votes
4answers
233 views

Is it possible to find generating functions of infinite sequences with Mathematica?

I'm trying to find the generating function of a sequence as $(0,1,0,1,0,1,\dots)$ but reading Mathematica's help on FindGeneratingFunction[] seems to tell me that ...
8
votes
2answers
215 views

Error computing sum of sum of digits

I've defined a function that computes the sum of the base-b digits of n: DigitSum[n_, b_] := Total[IntegerDigits[n, b]] Then I defined a function that computes ...
7
votes
1answer
94 views

Sum over Binomials and Gammas

Given the function, ...
7
votes
1answer
227 views

Understanding Dirichlet regularization in Sum

I've tried to calculate few classic sums using Dirichlet regularization: ...
7
votes
0answers
173 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 ...
6
votes
5answers
2k views

How to sum over a List

list = {11.5575, 11.397, 5.52734, 4.0878, 2.54815, 1.86652, 2.55028, 2.14952, 1.6242, 1.34117} I have a list of numbers. How do I make a function that creates a ...
6
votes
4answers
201 views

Define Function with Sum over a list

I want to define a function that would symbolically look like $$ t(s,\underline{a})=\pi s + \sum_{n=1}^{n_{max}}a_n\sin(n \pi s) $$ (something like a finite Fourier series). Here $s\in [0,1]$ and ...
6
votes
2answers
104 views

Sum considers RandomInteger[] as a constant

Perhaps this is expected behavior, but I was kind of surprised by the following: ...
6
votes
1answer
292 views

Summa package problem

I need to use the features of the Summa package but it doesn't work in Mathematica 9. It gives the error Cannot open Utilities`FilterOptions` My question is: ...
6
votes
3answers
377 views

Numerical evaluation of a sum

I am trying to compute numerically NSum[(-1)^n/n^3, {n, 1, Infinity}]. Of course, using first Sum would work here, but often ...
5
votes
3answers
262 views

The speed of Sum[] varies strangely

I was curious about the difference in speed between Total and Sum. I found out Total was ...
5
votes
2answers
165 views

Speed up plot of $\sum_{j\ge1} 2^{-j}(1-2^{-j})^{n-1}$

I'm a beginner at Mathematica. I would like to plot the following function: $${n\over2} \sum_{j\ge1} 2^{-j}(1-2^{-j})^{n-1}$$ However the following code is just too slow: ...
5
votes
3answers
168 views

Can I regroup terms into a sum?

Is there a way to regroup terms into a sum? I mean, for example, if I have the sequence $\quad \quad S_1 S_2 + S_2 S_3 + S_3 S_4 + S_4 S_5$ I would like to get the form $\quad \quad {\rm ...
5
votes
1answer
375 views

Problem with creating a large list of tuples

This is a follow-up question from Sum of Multinomial Coefficients I have thought about the meaning of the formula I mentioned and, with help, I implemented the following code: ...
5
votes
1answer
182 views

Compile nested Sums

I want to compile an expressions that contains nested Sum-expressions. A simple example that gives me problems is: ...
5
votes
1answer
219 views

Summing tensors in mathematica

How do I perform the following summation in mathematica? \begin{equation} \Sigma_{m=1}^5 e_{ijklm}A^{mn} \end{equation} I have the $e_{ijklm}$ tensor of rank 5 in 5 dimension as a array and $A^{mn}$ ...
5
votes
1answer
333 views

Variable number of nested variable-range sums

I would like to express the following nested sum in Mathematica: $$ S(m,j,N) = \sum_{k_1=m+j-1}^{N-1} f(N,k_1) \sum_{k_2=m+j-2}^{k_1-1} f(k_1,k_2) \cdots \sum_{k_m=j}^{k_{m-1}-1} f(k_{m-1},k_m) $$ ...
5
votes
2answers
572 views

Problems with Symbolic summation over unknown values

I'm having some real trouble with Mathematica wrongly evaluating various symbolic sums at the moment. I have this function: $$h_{ij}(x) = ...
5
votes
0answers
60 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 ...
5
votes
2answers
123 views

Simplification due to recognition of dummy indices in sums?

Today I noticed something weird concerning implicit sums. Apparently, Mathematica cannot recognize dummy indices as such and simplify accordingly. Consider: ...
4
votes
2answers
254 views

What causes this strange convergent sum?

N[Sum[1/(x^2 + 1), {x, 1, Infinity}], 5] N[Sum[1/(x^2 + x + 1), {x, 1, Infinity}], 5] 1.0767 0.79815 + 0.*10^-6 I What causes the strange number?
4
votes
4answers
244 views

How to stop a summation when a variable is small enough?

I meet with a problem. I hope to get an infinite summation of $f1(x)/f2(x)$ which converges to zero. So my code is Sum[f1[x]/f2[x],{x,Infinity}] or ...
4
votes
3answers
407 views

Why the difference?

When I do the double sum using the sigma notation I get $$1 + \sum_{n=0}^{\infty}\sum_{k = n}^{\infty} \frac{1}{(k+2)k!}$$ $1 + e - \cosh[1]$ When I do the sums as below, I get the expected ...
4
votes
2answers
172 views

Determine the coefficient of expansion of the product of two sumations?

I would like to determine the coefficient of a desired term in the product of two summation where the powers of $x$ are not necessarily integers. For example $$ \sum_{i=1}^N x^{1/i}\sum_{j=1}^N ...
4
votes
1answer
228 views

Summation with jump indices like 1,4,6,9?

I is easy to enter $ \sum_{s=1}^{n}k(s) $ But how can I enter $ \sum_{s=1,3}k(s) $ ? When I try it, Mathematica says ...
4
votes
3answers
208 views

Long waiting time for computing a summation

It takes a long time to compute the summation below, and I'd like to know if there are alternative ways to compute things faster. When replacing $15$ by $\infty$, then I should get $3^{1/3}$. I need ...
4
votes
2answers
214 views

Numerical sum does not give consistent results

Consider the function ...
4
votes
1answer
421 views

Von Mangoldt function

Can anybody evaluate the following sum for me $$ \sum\limits_{n=2}^\infty(-1)^n\left(\frac{\psi(n)}{n}-\frac{\Lambda(n)}{2n}\right) $$ where $\psi(n)$ is the Chebyshev function and $\Lambda(n)$ is ...
4
votes
2answers
1k views

Can I define a function for vectors of arbitrary dimension?

Is it possible to do analytic calculations with Mathematica? For example, I want to compute: $$\partial \frac{\sum_{j=1}^n G_{j} \prod_{k=1}^{j-1} (1 - G_{k})}{\partial G_l}=-\prod_{k\neq l} ...
4
votes
3answers
466 views

Summation with constraints

I am trying to do the summation shown below, $\sum_{i_1=0}^{imax_1} \sum_{i_2=0}^{imax_2} \dots \sum_{i_k=0}^{imax_k} f(i_1,\dots,i_k)$ $k$ is a variable. Therefore $i$ and $imax$ are defined as ...
4
votes
2answers
88 views

The fastest way of raising indices of high rank tensor

I have some high dimensional high rank tensors, let's say $$F_{ijkl}$$ and I need to find $$F^{abcd}=g^{ai}g^{bj}g^{ck}g^{dl}F_{ijkl}.$$ Here $g^{ij}$ is the contravariant metric. Simple summation ...
4
votes
1answer
412 views

How to sum over primes

Apologies in advance for the simplicity of the question, but I can't fathom how to write the following as a sum in Mathemaitca: \begin{align} &\sum_{p}^{a}\sum_{n}^{b}\text{expression} ...
4
votes
1answer
119 views

Converting a sum into $\Sigma$ notation

I have a simple expression of the form $\quad \quad t^5+t^4+t^3+t^2+t+1$ and I want Mathematica to convert this to the form $\quad \quad \sum _{i=0}^5 t^i$ Is there a way to do this?
4
votes
2answers
162 views

Solve a system of M equations without specifying M

This post has been edited heavily in response to the comments: my initial question was very confusing. I am trying to write my game-theoretic model in Wolfram Language. There is only one remaining ...