Let $W_1,W_2, \dots$ be the arrival times of a Poisson process $\{N(t) : t \geq 0\}$ of rate $\lambda$. Calculate $$E\left[\frac{W_{N(t)+1}}{N(t) + 1}\right].$$
My attempt: Let $S_n$ be the sojourn time between the $n$-th and $(n+1)$-st arrivals, then \begin{align} E\left[\frac{W_{N(t)+1}}{N(t) + 1}\right] &= \sum_{n=0}^\infty E\left[\left.\frac{W_{N(t)+1}}{N(t) + 1}\right|N(t)=n\right]P(N(t)=n)\\ &=\sum_{n=0}^\infty E\left[\left.\frac{W_{n+1}}{n + 1}\right|N(t)=n\right]P(N(t)=n)\\ &=\sum_{n=0}^\infty E\left[\left.\frac{W_{n}+S_n}{n + 1}\right|N(t)=n\right]P(N(t)=n) \end{align} Conditioning on $N(t)=n$, $W_n=\mathrm{max}\{U_1,\dots,U_n\}$ hence $E[W_n|N(t)=n]= tn/(n+1)$. Due to independence $E[S_n|N(t)=n] = E[S_n]=1/\lambda$, so the sum evaluates to \begin{align} \sum_{n=0}^\infty E\left[\left.\frac{W_{n}+S_n}{n + 1}\right|N(t)=n\right]P(N(t)=n) &= \sum_{n=0}^\infty\frac{1}{n+1}\left(\frac{tn}{n+1}+\frac{1}{\lambda}\right)\frac{e^{-\lambda t}(\lambda t)^n}{n!}\\ &=\sum_{n=0}^\infty\frac{tne^{-\lambda t}(\lambda t)^n}{(n+1)!(n+1)}+\frac{e^{-\lambda t}(\lambda t)^n}{\lambda(n+1)!}. \end{align}
I don't know if there are any flaws in my argument leading up to the wrong calculation but I have been stuck trying to simplify the first summand. If anyone can point me in the right direction it would be great, thanks!