MATLAB: Infinit symbolic series with symsum() – HELP!

Question

Hello everyone,

I am trying to solve the following infinite series problem at MATLAB R2015a:

I have scripted the following:

clear all
clc
syms x t a n
N = [n, n+2]; %Vector with n and n+2
A_N = ((tan(x)).^N);%Integrant of a_n
a_n = int(A_N,x,0,pi/4);%Integral term a_n
s = (1/n)*(a_n(1) + a_n(2));%General term of the series
S = symsum(s,n,1,inf);

which yields for the varialbe S:

S = symsum((int(tan(x)^n, x, 0, pi/4) + int(tan(x)^(n + 2), x, 0, pi/4))/n, n, 1, Inf)

If I use vpa(S), it yields: ans = 1.0

What am I doing wrong? How could I solve this problem properly in matlab?

Thank you in advance

Gabriel

Answer 1

You did make an effort, even though this is homework. And you even got the correct answer. But pencil and paper should suffice, maybe with just a little help from MATLAB. Suppose you make the obvious transformation? That is, what if

u = tan(x)

then we have dx = 1/(u^2 + 1) du. Can you now write a_n in simple form? Perhaps this does not get you all the way there, at least not without a little more work.

Now consider what happens if you form the sum of a_n + a_(n+2). Does that help any? (Hint: YES.) That is, what if you combine

a_n + a_(n+2)

as an integral? Does anything nice happen? Can you factor anything? Hint:

(u^n + u^(n+2))/(u^2 + 1) == u^n

So can you compute a_n + a_(n+2)? I hope so, but to make you feel better, I'll use MATLAB. ;-)

syms u
syms n real positive
int(u^n,[0,1])
ans =
1/(n + 1)

Now, what is the infinite sum?

symsum(1/n/(n+1),1,inf)
ans =
1

What does this say? That you got the correct answer, yet did not believe it would be that simple? Sometimes homework questions really do have a simple solution. In fact, that would be the perfect homework problem - making you do some work, only to yield a simple answer. Sadly, real life is rarely that nice, so enjoy being a student while it is there.