How to show that $e^{x+y} = e^x e^y$ by series expansion I know that $e^xe^y=e^{x+y}$ but I want to show it by expanding the exponentials in MacLaurin Series.
$$
\left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} \frac{y^m}{m!}\right) =^? \sum_{n=0}^{\infty} \frac{(x+y)^n}{n!}
$$
This is what I have.  I'm not sure what my next step would be since I can't think of a way to combine the two summations on the RHS.

Hint: look at the Wikipedia article on the Cauchy product. In general, if $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ both converge, and at least one converges absolutely, we have
$$\left(\sum_{n=0}^\infty a_n\right)\left(\sum_{n=0}^\infty b_n\right)=\sum_{n=0}^\infty\left(\sum_{k=0}^n a_{n-k}b_k\right).$$
Apply this to the product of series you are considering, and be sure to use the binomial theorem.
