Consider the integral: ∫01sin⁡(πx)1−xdx I want to do this via power series and obtain an...



Answered question


Consider the integral:
I want to do this via power series and obtain an exact solution.
In power series, I have
My question is: how do I multiply these summations together? I have searched online, however, in all cases I found they simply truncated the series and found an approximation.

Answer & Explanation



Beginner2021-11-20Added 16 answers

Let's take a more abstract case, trying to multiply {k=0}an  and  {k=0}bn. Note that In the resulting sum, we will have aibj for all possibilities of i,j N.
One way to make it compact is to sum across diagonals. Think about an integer lattice in the first quadrant of R2. Drawing diagonals (origin, then along x+y=1 then along x+y=2, etc), note that the one along the line x+y=n will have length n+1 integer points, and the sum of the indices along all points there will be n - i.e.
(n,0),(n−1,1),…,(k,n−k)…,(0,n). So we can renumber the summation based on these diagonals, getting
(k=0an)(k=0bn)={n=0}j,k,along  x+y=nakbj={n=0}{k=0}akbnk


Beginner2021-11-21Added 15 answers

I am trying to solve and it does not work, if you can, then please help

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?