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

Let's take a more abstract case, trying to multiply ${\displaystyle \sum _{\{k=0\}}^{\mathrm{\infty }}{a}_n\text{ and }\sum _{\{k=0\}}^{\mathrm{\infty }}{b}_n}$. Note that In the resulting sum, we will have ${\displaystyle a_i{b}_j}$ for all possibilities of i,j ${\displaystyle \in \mathbb{N}}$.
One way to make it compact is to sum across diagonals. Think about an integer lattice in the first quadrant of ${\displaystyle {\mathbb{R}}^2}$. 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
${\displaystyle \left(\sum _{k=0}^{\mathrm{\infty }}{a}_n\right)\left(\sum _{k=0}^{\mathrm{\infty }}{b}_n\right)=\sum _{\{n=0\}}^{\mathrm{\infty }}\sum _{j,k,\text{along }x+y=n}{a}_k{b}_j=\sum _{\{n=0\}}^{\mathrm{\infty }}\sum _{\{k=0\}}^{\mathrm{\infty }}{a}_k{b}_{n-k}}$

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