Let $f(x)=\sum_{i=0}^{m} a_i x^i$ and $g(x)=\sum_{i=1}^{n} b_i x^i$ be two polynomial then their product is defined by $(fg)(x)=\sum_{k=0}^{m+n} a_kb_{k-i} x^{k} $.
Now if we tend $m,n \to \infty $ and let both the series $f$ and $g$ converges on $\mathbb{R}$, then how to write their product. i.e., $ \left( \sum_{i=0}^{\infty} a_ix^i \right) \left(\sum_{i=1}^{\infty} b_i x^{i} \right)=? $
Can we express the product of the two series just similar form as in case of polynomial?
Help me, please.
On
If $f(x)=\sum_{n=0}^{\infty} a_n x^n$ and $g(x)=\sum_{n=0}^{\infty} b_n x^n$ are power series which are convergent for $|x|<R$, then we have
$f(x)g(x)=\sum_{n=0}^{\infty} c_n x^n$ for $|x|<R$,
where $c_n= \sum_{k=0}^{n} a_kb_{n-k}$.
The power series $\sum_{n=0}^{\infty}c_nx^n$ is called the Cauchy product of $\sum_{n=0}^{\infty}a_nx^n$ and $\sum_{n=0}^{\infty}b_nx^n$
The classical theorem is the following :
If $\sum_{n=0}^{\infty} u_n$ and $\sum_{n=0}^{\infty} v_n$ are two series that converge absolutely, then the series $\sum_{n=0}^{\infty} w_n$, where $w_n = \sum_{k=0}^n u_k v_{n-k}$, also converge absolutely and we have $$\sum_{n=0}^{\infty} w_n = \left( \sum_{n=0}^{\infty} u_n \right)\left( \sum_{n=0}^{\infty} v_n \right)$$