let all xn>=0 for all natural numbers n. Assume that sigma(n from 1 to infinity) xn<infinity
Prove that the lim as n tends to infinity of xn=0
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.