how to multiply infinite power series I am doing an assignment for my precalculus 2 class.  I am expanding two infinite power series and multiplying them together to prove that $\exp(ax)\exp(by) = \exp(ax+by)$  I'm not sure what I am supposed to do when I have $x(...)$ or $x+(...)$  should $(...)$ be the summation of all terms between the terms surrounding it?  I know that that is technically what it means but I'm not sure how to operate on it.  


If you have two power series $A(x)=\sum_{n\ge 0}A_nx^n$ and $B(x)=\sum_{n\ge 0}B_nx^n$, then their (formal) product is given by the power series $C(x)=A(x)B(x)=\sum_{n\ge 0}C_nx^n$, whereby
$C_n=\sum_{k=0}^n A_kB_{n-k}$
This is sometimes called Cauchy's product formula but you may want to verify this yourself. Hence, for the product of $\exp(ax)$ and $\exp(bx)$, this gives
$C_n = \sum_{k=0}^n \frac{a^k}{k!}\frac{b^{n-k}}{(n-k)!}$.
Multiply by $n!$ and obtain
$n!C_n = \sum_{k=0}^n \binom{n}{k}a^kb^{n-k}=(a+b)^n$.
Therefore:
$\exp(ax)\exp(bx)=\sum_{n\ge 0}\frac{(a+b)^n}{n!}x^n = \exp((a+b)x)$.
