Let $X_1,X_2...$ be jointly independently random variables with common pdf given by $f(x_i)$. I am trying to show that $E[N]$ is infinite, where $N$ is the number of years till $X_1$ is exceeded for the first time, and $X_i$ represents the annual rainfall in the i-th year.
First I obtained the distribution function for N. Since $F(X_1)$ is the probability of a particular observation being less than $X_1$, it can be shown that: $$p(N=n)=F(X_1)^{n-1}(1-F(X_1)).$$
Next, I can express the expected value of $N$ as: $$E[N]=\sum_1^\infty nF(X_1)^{n-1}(1-F(X_1))=\sum_1^\infty nF(X_1)^{n-1} - \sum_1^\infty nF(X_1)^{n}. $$
At this point, I'm not sure how to proceed. Any advice?
The probability that $N\ge n$ is the probability that $X_1$ is the greatest of the first $n$ values, which is $\frac1n$. Thus
$$ \mathsf E[N]=\sum_{n=1}^\infty\mathsf P(N\ge n)=\sum_{n=1}^\infty\frac1n=\lim_{k\to\infty}H_k=\infty\;. $$