I know how to determine what the series $\displaystyle \sum_{n=1}^{\infty}n^pr^{n-1}$ converge to for positive integral values of p:

p = 1: $\displaystyle \frac{1}{(1 - r)^2}$

p = 2: $\displaystyle \frac{1+r}{(1 - r)^3}$

p = 3: $\displaystyle \frac{1 + 4r + r^2}{(1 - r)^4}$

p = 4: $\displaystyle \frac{1 + 11r + 11r^2 + r^3}{(1 - r)^5}$

p = 5: $\displaystyle \frac{1 + 26r + 66r^2 + 26r^3 + r^4}{(1 - r)^6}$

The general pattern is:

$\displaystyle \frac{a_0 + a_1r + ... + a_{p-2}r^{p-2} + a_{p-1}r^{p-1}}{(1 - r)^{p+1}}$

where the coefficients $\displaystyle a_0, a_1, ... a_{p-1}$ are given by the pth row of Euler's triangle (there is an article on the triangle at

www.mathworld.com).

So I figured this out but I don't know how to prove it formally. Any suggestions?