Let $X_1,X_2,...$ be independent random variables with a given continuous distribution function $F(x)=\mathbb P(X_i\le x), x>0$ and $F(0)=0$. The values $X_1,...,X_n$ have been recorded.
Let $Y_n=\operatorname{max}[X_1,...,X_n]$.
Let $\tau=\operatorname{min}[t>1:X_{n+t}>Y_n]$
Find $\mathbb E(\tau)$
We have shown that $\mathbb P( \tau=k)=\frac n{(k+n-1)(k+n)}$ but surely we then have $$\mathbb E(\tau)=\sum^{\infty}_{k=1}k\space\mathbb P(\tau=k)=\sum^{\infty}_{k=1}\frac{kn}{(k+n-1)(k+n)}$$ but now in the case $n=1$ (and the question gives no indication that we cannot consider $n=1$) don't we have: $$\mathbb E(\tau)=\sum^{\infty}_{k=1}\frac 1{k+1}?$$
This must be wrong but I don't see where.
Any help with the question or where I've gone wrong would be greatly appreciated. Thanks