# Items tagged with summationsummation Tagged Items Feed

### Euler-Maclaurin Summation...

April 06 2014
0 1

Greetings to all.

I would like to share a brief observation concerning my experiences with the Euler-Maclaurin summation routine in Maple 17 (X86 64 LINUX). The following Math StackExchange Link shows how to compute a certain Euler-MacLaurin type asymptotic expansion using highly unorthodox divergent series summation techniques. The result that was obtained matches the output from eulermac which is definitely good to know. What follows is the output from said routine.

> eulermac(1/(1+k/n),k=0..n,18);
1       929569        3202291        691                O(1)
O(- ---) - ----------- + ----------- - --------- + 1/1048576 ----
19             15            17          11              19
n      2097152 n     1048576 n     32768 n               n

n
/
174611      5461        31       |      1           17        1
- -------- + --------- + ------- +  |   ------- dk - ------- + ------
19          13         9    |   1 + k/n            7        5
6600 n     65536 n     4096 n    /                 4096 n    256 n
0

1       1
- ------ + ---- + 3/4
3   16 n
128 n


While I realize that this is good enough for most purposes I have two minor issues.

• One could certainly evaluate the integral without leaving it to the user to force evaluation with the AllSolutions option. One can and should make use of what is known about n and k. In particular one can check whether there are singularities on the integration path because we know the range of k/n.
• Why are there two order terms for the order of the remainder term? There should be at most one and a coefficient times an O(1) term makes little sense as the coefficient would be absorbed.

You might want to fix these so that the output looks a bit more professional which does enter into play when potential future users decide on what CAS to commit to. Other than that it is a very useful routine even for certain harmonic sum computations where one can use Euler-Maclaurin to verify results.

Best regards,

Marko Riedel

### How do I apply an inverse distributive law...

January 02 2014
0 2

I have the following expression (obtained from an earlier calculation):

I want to collect all the terms under one summation. So I define a rule:

collectf:=proc(f)
A::algebraic*f(a::algebraic)+B::algebraic*f(b::algebraic)\
+C::algebraic*f(c::algebraic)+D::algebraic*f(d::algebraic)=f(A*a+B*b+C*c+D*d);
end proc:

and then

applyrule(collectf(Sum),%);

I get

Error, (in +) unable to identify A::algebraic

I used similar constructs before so I think the rule is constructed correctly. I should, however, mention that I use the Physics:-Vectors package and in fact the expression I start up with here reads, in 1-d Maple inputform:

Physics[Vectors][+](Physics[Vectors][+](Physics[Vectors][+](-y*(Sum((diff(a[n](r), r))/(exp(I*Pi*n/L))^2, n))/r, (2*I)*(Sum(a[n](r)/(exp(I*Pi*n/L))^2, n))*k0), y*(Sum(a[n](r)/(exp(I*Pi*n/L))^2, n))*k0^2), -y*(Sum((diff(a[n](r), r, r))/(exp(I*Pi*n/L))^2, n)))

Is my problem related to the use of Physics:-Vectors? If so, how can I get around that?

TIA,

Mac Dude

### display of summation of numbers as the outcome...

January 02 2014
1 6

How can I show the expression of the following summation as the output, not 11?

3+7+1

### Physics updates and multi-index summation...

December 05 2013
2 0

Hi,
Relevant developments in Physics happened during the last month and a 1/2, some of them of interest beyond the use of this package. Among the most exciting things I can mention:

1. The redefinition of the derivative rule for the complex components (abs, argument, conjugate, Im, Re, signum) together with the introduction of Wirtinger calculus, as an user-option to work with complex variables. In other words: it is now possible to compute taking z and its conjugate as independent variables and in equal footing.
2. Introduction of textbook mathematical display for all the inert functions of the mathematical language, also for unknown functions f(x).
3. New options in Physics:-Setup to indicate that some mathematical objects are real (different from assume(x, real), while integrated with is and coulditbe).
4. A rather large number of micro developments advancing the integration of Physics with simplify, expand and combine.
5. Another large number of micro developments for quantum mechanics.
6. New options in Physics:-Setup to redefine sum as Physics:-Library:-Add, and with that have access to multiindex summation directly from sum, for instance as in sum(f(i, j), i + j <= n), including related typesetting copy & paste.

As usual the latest version of the package is available for download in the Maplesoft Physics: Research & Development webpage  and in the zip there is a worksheet illustrating all these developments. Below I'm copying the section related to the new redefinesum option of Physics:-Setup and multiindex summation.

Thanks to everyone who provided feedback, it has been of great value and at the root of this new round of developments.

December 4

 • New option in Setup: redefinesum, so that the sum command is redefined in such a way that     a) the sum arguments are processed in a way avoiding premature evaluation and related unexpected results or error interruptions     b) the sum command includes new functionality present in Physics:-Library:-Add to perform sum over integer values of many indices, as in

or

 >
 (1.1)

New option: redefine sum so that its arguments are processed by the more modern Physics:-Library:-Add and so that it can perform multiindice summation.

Example:

By default, the sum command is not redefined, so the value of redefinesum is

 >
 (1.2)

Consider this multiindex summation functionality of the Physics:-Library:-Add command

 >
 (1.3)

For instance, for n = 2,

 >
 (1.4)

This functionality can be plugged directly into the sum command. For that purpose, set redefinesum to true

 >
 (1.5)

You can now compute directly with sum. The left-hand side is inert while the right-hand side is computed

 >
 (1.6)
 >
 (1.7)
 >
 (1.8)

The formula for the integer power of a sum

 >
 (1.9)
 >
 (1.10)
 >
 (1.11)

Verify whether this equation is true

 >
 (1.12)

Besides this new functionality, the redefined sum does a more modern handling of its arguments, consider a typical problem posted in Maple primes

 >
 (1.13)

In the following summation, j is a dummy summation index, so the value just assigned, , is not expected to interfer with the summation. This is the case with the redefined sum

 >
 (1.14)

while without redefining sum the input above is interrupted with an error message. Likely, in this other case also reported in Mapleprimes

 >
 (1.15)

the following two summations can be performed after having redefining sum:

 >
 (1.16)

For the summation above, without redefining sum, it returns 0 instead of unevaluated, because of a premature evaluation of the function  with an unassigned index i before performing the summation. Returning unevaluated as (1.16) permits evaluate the sum at a latter moment, for instance attributing a value to f

 >
 (1.17)

And this other sum where f is given from the begining also returns 0 without redefining sum

 >
 (1.18)

Problems like this other one reported in Mapleprimes here also get resolved with this redefinition of sum.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

### Find integers A and B please!...

November 18 2013
1 3

Would really appreciate the help in doing this question!

Pi= infinity Σ n=0 ((120n^2+An+B)/(16^n(512n^4+1024n^3+712n^2+194n+15))) for some positive integers A,B.

Hint: Set A=0 to find B, then find A.

I'm so clueless as to how to do this. THANK YOU!

### print a summation of binomial coefficients...

September 09 2013
0 1

Hi, I want to print     t[n] / (-1)^n * sum   as following, but I don't know where the error is. Please help me.

Thank you very much.

### How to convert this into an expression of Summatio...

August 30 2013
0 1

convert 1/(1+z) to

Summation(....

### Simga (Could you do more clearly ?)...

August 26 2013
0 1

Let $n$ be any positive Integer :$\displaystyle \sum_{k=0}^{n-1}(\sqrt{ak^3+bk^2+ck+1}-\sqrt{ak^3+bk^2+ck})^{\frac{1}{3} }=\sqrt{n}$. Find

### How to include constant variables in summation and...

August 11 2013
1 2

Dear all,

I'm trying to substitute a combination of variables in an equation that is partly covered in a summation.

For example consider the equation with two variables A and B, and two function W(t) and q(z):

now I want to do the substitution:

### gradient in terms of sum set of indexed variables ...

June 26 2013
0 0

How does one take the gradient of a expression involving sums in terms of a set of indexed variables.

I am trying to find the maxima for the satisfaction for a public good from a tax code. The sum represents the total revenue. The tax code is defined as a piecewise linear function of income and indexed variables. The indexed variables represent the points of the kinks and the marginal tax rate for each income bracket. (The set of indexed variables will vary...

### help in write a summation on Maple...

May 13 2013
1 2

Hi

I want to write the following summation but Maple give the following error

summation:

### Physics: can i differentiate expression with prese...

March 21 2013
1 3

Hi there,

I have

x[i]=sum(sin(theta[j](t)), j = 1 .. i-1)

assuming i is integer and  i>=1 can Maple help me to differentiate with respect to time t ( for example diff(x[i],t) ) so

### How to summation and simplify q calculus wtih QDif...

February 17 2013
0 0

i am doing a Q Laplace and finally summation it

however, i find an AccurateQSummation which is not like Summation

Should i use it, if not, use SumTools' summation how to simplify it?

restart;with(QDifferenceEquations):qexp := QPochhammer(-(-p*t), q, infinity);f:= qexp*t;b := subs(t=q^n,f);QSimplify((1-q)*AccurateQSummation(b*q^n,n=-infinity..infinity));with(SumTools):QSimplify((1-q)*Summation(b*q^n,n=-infinity..infinity));

### How to laplace transform for hypergeometric form i...

December 21 2012
0 1

How to laplace transform for hypergeometric form in maple

### What do rsolve solve for...

December 21 2012
0 0

if rsolve is solving difference equation for L(x) in summation(L*z^n, n=0..infinity)

can i use double encapsulation to solve for summation(L*z^n/n!, n=0..finity)

step 1 use rsolved result of a given classic difference equation times z^n/n! * t^n

step 2 then summation step 1 and use celine method to change into difference equation again

step 3 solve this new difference equation

then i imagine L should be L*z^n/n!

but i am not sure...

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