Agaiepsh

2021-11-19

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.

Drood1980

Beginner2021-11-20Added 16 answers

Let's take a more abstract case, trying to multiply $\sum _{\{k=0\}}^{\mathrm{\infty}}{a}_{n}\text{}\text{and}\text{}\sum _{\{k=0\}}^{\mathrm{\infty}}{b}_{n}$ . Note that In the resulting sum, we will have $a}_{i}{b}_{j$ for all possibilities of i,j $\in \mathbb{N}$ .

One way to make it compact is to sum across diagonals. Think about an integer lattice in the first quadrant of$\mathbb{R}}^{2$ . 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

$\left(\sum _{k=0}^{\mathrm{\infty}}{a}_{n}\right)\left(\sum _{k=0}^{\mathrm{\infty}}{b}_{n}\right)=\sum _{\{n=0\}}^{\mathrm{\infty}}\sum _{j,k,\text{along}\text{}x+y=n}{a}_{k}{b}_{j}=\sum _{\{n=0\}}^{\mathrm{\infty}}\sum _{\{k=0\}}^{\mathrm{\infty}}{a}_{k}{b}_{n-k}$

One way to make it compact is to sum across diagonals. Think about an integer lattice in the first quadrant of

(n,0),(n−1,1),…,(k,n−k)…,(0,n). So we can renumber the summation based on these diagonals, getting

Onlaceing

Beginner2021-11-21Added 15 answers

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