Yes, it is 1/3. Given a series of partial sums (which in turn form an infinite sequence), then if the infinite sequence converges to some number B, the infinite

Yes, it is 1/3. Given a series of partial sums (which in turn form an infinite sequence), then if the infinite sequence converges to some number B, the infinite sum likewise converges to the same number B. It's exactly like saying lim as x -> 1 of 2x -> 2. You don't write it that way. You write it as lim as x -> 1 of 2x = 2.