I have this scenario where $S[n] = 0$ if an email does not arrive and $S[n] = 1$ if an email arrives in the $n^{th}$ minute. Therefore, $S[n]$ follows a Bernoulli distribution. Now $U[m]$ is the number of minutes between the $m^{th}$ and $(m+1)^{th}$ email where $m = 0, 1, 2, 3...$. Therefore, $U[m]$ follows a Geometric distribution. Also $P(S[n] = 0) = p$, therefore, I have concluded that,
$P_{U[m]}(m) = p^{m}(1-p)$
I am required to determine the autocorrelation of $U[m]$, and so far, I have:
$R_{UU}[m, m+p] = E[U[m]] \times E[U[m+p]] = \frac{p^2}{(1-p)^2}$ for $p \ne 0$
But for $p = 0$, I need to find:
$E[U^2[m]] = \sum_{m=0}^{\infty}u_m^2 \cdot P_{U[m]}(m)$
Which is also:
$E[U^2[m]] = \sum_{m=0}^{\infty}m^2 \cdot (1-p) \cdot p^m$
$E[U^2[m]] = (1-p) \cdot p \sum_{m=0}^{\infty}m^2 \cdot p^{m-1}$
Now I am not sure how to proceed?