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

Agaiepsh 2021-11-19 Answered
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.
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Expert Answer

Drood1980
Answered 2021-11-20 Author has 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
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Onlaceing
Answered 2021-11-21 Author has 15 answers
I am trying to solve and it does not work, if you can, then please help
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