I am trying to find the -1 moments of sum of N geometric random variable, i.e. $E[\frac{1}{\sum_{i=1}^N X_i}]$

Suppose the probability mass function is $f_X(x) = (1 - p)^{x - 1} p$

The moment generating function of $\sum_{i = 1}^N X_i$ will be

$E[e^{t \sum X_i}] = ( \frac{pe^t}{1 - (1-p)e^t})^N$

Let $h(t) = \frac{pe^t}{1 - (1-p)e^t}$, the -1 moments of $\sum_{i = 1}^N X_i$, will just be $\int h(t)^N dt |_{t = 0}$. This is where I got stuck....