I have a bounded, integer values random variable $X_n$ which lies between $0$ and $n$. I want to find the expectation of $\frac{1}{X_n+1}$.
I want to say that $E[\frac{1}{X_n+1}] = \sum_{j=1}^\infty (-1)^jE[X_n^j]$, but I am worried about the convergence of this.
$\frac{1}{x+1}$ converges for $|x| < 1$, but it is not true that $E[X_n^j] = E[X_n]^j$, so I am unable to determine if finite moments is enough for convergence of the expectation of the fraction in question
Using the equation $1/(1+X)=\int_0^1 s^X\,ds$, we have \begin{align} E\left[\frac1{X+1}\right] &=E\left[\int_0^1s^X\,ds\right] \\&=E\left[\int_0^1(1+(s-1))^X\,ds\right] \\&=E\left[\int_0^1\sum_{k=0}^n(s-1)^k\binom{X}k\,ds\right] \\&=\sum_{k=0}^nE\left[\binom{X}k\right]\int_0^1(s-1)^k\,ds \\&=\sum_{k=0}^nE\left[\binom{X}k\right]\frac{(-1)^{k}}{k+1} \end{align} Now, how is this helpful? Well, you mentioned that $X=Y_1+Y_2+\dots+Y_n$ is a sum of indicators. This means that $$ \binom{X}k=\sum_{1\le i_1< i_2< \dots< i_k\le n}Y_{i_1}Y_{i_2}\cdots Y_{i_k} $$ so $E[\binom{X}k]$ is the sum over all $k$-wise intersections of the probability that that intersection occurs. $$ E\left[\binom{X}k\right]=\sum_{1\le i_1< i_2< \dots< i_k\le n}P(Y_{i_1}=Y_{i_2}=\cdots =Y_{i_k}=1) $$ I imagine these probabilities are somewhat natural to compute in the context of your problem.