I'm having some confusion over this statement here. Let $T_i \sim Exp(\lambda + \theta)$ and if they are all iid then $\sum_n T_i \sim Gamma(\alpha = n, \beta = 1/(\lambda + \theta))$
I want to find $E(\frac{1}{\sum_n T_i})$. I know I can't just do the reciprocal of $E(\sum_n T_i)$
Do I integrate over $\int_0^\infty \frac{t}{f(t)} dt$?
If I do this I get $\Gamma(\alpha) \beta^\alpha \int_0^\infty \frac{e^{t/\beta}}{t^\alpha} dt$ and I don't know if this is the right way to proceed since I end up with incomplete gamma?
I know that I'm supposed to end up with $\frac{\lambda + \theta}{n-1}$ as the answer but I'm not sure how the gamma is supposed to cancel out.