TITLE: How can expected value be infinite?
QUESTION [7 upvotes]: My book as well as Wikipedia 
gives this definition of expected value:

$\mathbb E(X)=\sum _x xf(x).$ But, $\mathbb E(X)$ is said to exist if and only if that equation is absolute convergent. 

But, I see that many places do not follw this def., e.g. here $${\mathbb E} X = \sum_{n=1}^\infty 2^{-n} \cdot 2^n = \sum_{n=1}^\infty 1 = \infty,$$ though we can see that absolute convergence test is failed.
So, can expected value be infinity or negative infinity?

REPLY [2 votes]: I just answered a similar question on SE stats, CrossValidated.  Rather than copy that answer over here, I give the link:
https://stats.stackexchange.com/questions/94402/what-is-the-difference-between-finite-and-infinite-variance
That answer is much longer and more detailed than the ones above here!