Expectation; example of infinite sum?

Question

My book gives the following definition for expectation:

If $X$ is a discrete random variable, the expectation of $X$ is denoted by

$\mathbb{E}(X)$ and defined by $\mathbb{E}(X)=\sum_{x \in > Imx}x\mathbb{P}(X=x)$

whenever this sum converges absolutely, in that $\sum_{x}|{x\mathbb{P}(X=x)|<\infty}$.

Is there a practical example where we would have a divergent sum, so that $\mathbb{E}(X)$ is not defined?

I'm asking this to make it easier for me to remember this remark on infinite sums.

Answer 1

Take, for example, a disrete random variable $X$ given by $$\Bbb P(X=k) = \frac{6}{\pi^2 k^2},\quad k=1,2,3,\ldots$$

This is a valid random variable, yet its expectation does not exist, because the series $\sum_{k\ge 1} \frac 1k$ diverges.