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.  


Cauchy's product formula is easy to prove, if you look closely at what formulas with sums mean.
Take $A(z) = \sum_{r \ge 0} a_r z^r$ and $B(z) = \sum_{s \ge 0} b_s z^s$. Multiply:
\begin{align}
A(z) \cdot B(z) 
  &= \left( \sum_{r \ge 0} a_r z^r \right) \cdot \left( \sum_{s \ge 0} b_s z^s \right) \\
  &= \sum_{r \ge 0} \sum_{s \ge 0} a_r b_s z^{r +s} \\
  &= \sum_{n \ge 0} z^n \sum_{0 \le r \le n}a_r b_{n - r}
\end{align}
The first is just that $r$ and $s$ are totally independent, so you can distribute at will, the second is just grouping terms with $n = r + s$, thus $s = n - r$. The range on $r$ is just to limit to terms that make sense, no negative indices.
