Product of Infinite Series

Question

I am trying to compute the product of 3 infinite series.

As such, I need the compact form for the product $$\displaystyle\sum_{n=0}^{\infty}c_{n}\times\displaystyle\sum_{j=0}^{\infty}\sum_{k=0}^{j}a_{k}a_{j-k}.$$ I got the double summation by applying Cauchy Product.

Thanks, Radz.

Answer 1

To express it as a single infinite series you can apply the Cauchy Product a second time to get

$$ \sum_{m=0}^\infty e_m $$

where

$$ e_m = \sum_{l=0}^m c_l \sum_{k=0}^{m-l} a_kb_{m-l-k} $$

Answer 2

$$\sum_{n=0}^{\infty}a_{n}\times\sum_{n=0}^{\infty}b_{n}\times\sum_{n=0}^{\infty}c_{n}=\sum_{n=0}^{\infty}s_{n}$$ where $$s_{n}=\sum_{r+s+t=n}a_{r}b_{s}c_{t}$$