$(X_k)_k$ random variables sequence, iid with geo(p) distribuition.
Using this sequence propose an approximation of this sum:
$\sum_{k=0}^{ \infty } \sum_{j=0}^{k} p^2(1-p)^{k+j}j$
I was trying to do it somehow using SLLN but I got stuck trying to do something with the inner sum:
$\sum_{j=0}^{k} (1-p)^jj= (1-p)\left( \sum_{j=0}^{k}(1-p)^j\right)'=-\frac{k(1-p)^kp-1+(1-p)^k}{p^2}$
I don't know if I made a mistake with calculations or it is okay but I don't know what to do with this next.