Yet another way: Let be the nth partial sum. Then the series converges if and only if the sequence of partial sums converges (that's the definition of convergence of a sequence).
But then, if does not go to 0, does not go to 0. That means that cannot go to 0 as n and m go to 0 independently and so is not a Cauchy sequence.
Since every convergent sequence is a Cauchy sequence, does not converge and thus does not converge.