Sum of alternating series is non-zero?

The sum is always $\ge$ each even-numbered partial sum and $\le$ each odd-numbered one. So $S \ge 1 - 1/3 = 2/3 > 0$.

If a sequence converges to a limit, then every subsequence converges, to the same limit. In the case of an infinite convergent sum, this implies that the partial sums with an even number of terms will converge to the same value. However, those partial sums are of positive numbers such as $1-1/3$, $1/5-1/7$, $1/9-1/11$ etc. so they make up an increasing sequence of positive numbers, and so the limit must be positive.

The same applies to your general case.