I know how to determine what the series converge to (for -1 < r < 1) for positive integral values of p:

p = 1:

p = 2:

p = 3:

p = 4:

p = 5:

The general pattern is:

where the coefficients 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?