Let $X_1, X_2,\dots,X_n$ be $n$ i.i.d. geometric distributed random variables with successful probability $p$, that is $P(X_i=k)=(1-p)^{k-1}p$
Let $Y_i=X_i-\mathbb{E}(X_i) = X_i-\frac{1}{p}$, so $Y_1,\dots,Y_n$ are also i.i.d. random variables.
Let $S_k=Y_1+\dots+Y_k$, and $S_k^+=\max(0,S_k)$
I want to know a close form of $\mathbb{E}(S_k^+)$