Power series involving complex variable. Show that
$1 + \binom{m+1}{1}z + \binom{m+2}{2}z^2 +...+ \binom{m+n}{n}z^n +... = \frac{1}{(1-z)^{m+1}}$
for non-negative integers $m$ and $|z| < 1$.

Hint: start with the geometric series:
$$1+z+z^2+\ldots = \frac{1}{1-z}$$
and differentiate. For example,
$$1+2 z+3 z^2+\ldots=\frac{1}{(1-z)^2}$$
$$2+(3)(2)z+(4)(3)z^2+\ldots=\frac{2}{(1-z)^3} \implies 1+\binom{3}{1} z+\binom{4}{2}z^2+\ldots=\frac{1}{(1-z)^3}$$
Differentiate $m$ times to get your result.
