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?