Consider the integral: \int_0^1\frac{sin(\pi x)}{1-x}dx I want to do this via power

Agaiepsh

Agaiepsh

Answered question

2021-11-19

Consider the integral:
01sin(πx)1xdx
I want to do this via power series and obtain an exact solution.
In power series, I have
01({n=0}(1)n(πx)2n+1(2n+1)!{n=0})dx
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

Drood1980

Drood1980

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
Onlaceing

Onlaceing

Beginner2021-11-21Added 15 answers

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

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