Consider the integral:

${\int}_{0}^{1}\frac{\mathrm{sin}\left(\pi x\right)}{1-x}dx$

I want to do this via power series and obtain an exact solution.

In power series, I have

${\int}_{0}^{1}(\sum _{\{n=0\}}^{\mathrm{\infty}}{(-1)}^{n}\frac{{\left(\pi x\right)}^{2n+1}}{(2n+1)!}\cdot \sum _{\{n=0\}}^{\mathrm{\infty}})dx$

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.

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.