I am on the fence about whether this is a good troll or a bad troll. It certainly exploits the "someone on the internet is WRONG" ethos of HN, but I don't think it does so in a particularly amusing way. I'm going to say that it's a rather boring troll.
(I do find myself compelled to say that the mathematical convention is that an infinite sum is defined to be equal to the limit of the partial sums, if it exists.)
Considering when you created your account, and the subject of your post, "troll" seemed like the most likely explanation. Sorry about that. (It's sort of a typical troll to make a technically incorrect statement and then watch people get all agitated as they correct you. In fact, it's one of my favorite trolls, right behind making a sarcastic statement which you know will be taken literally by half of the audience, a la "A Modest Proposal".)
Anyways, assuming that your comment was in earnest, the arrow is typically used for functions (or sequences). E.g.,
1/n -> 0 as n->infinity
You could write:
1/4 + 1/16 + ... + 1/4^n -> 1/3
But you would write
1/4 + 1/16 + ... = 1/3
You wouldn't write (or at least I've never seen it)
1/4 + 1/16 + ... -> 1/3
It's not really a mathematical issue, just a definitional one: the left hand side is considered a real number, not a sequence of real numbers (or function :N->R, or whatever).
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.
1/3 is the limit, as the sum of n=1 to n -> infinity, of (1/4)^n
The "result" converges towards 1/3. You can get as close to 1/3 as you like, but the result will never quite equal 1/3.
Cheers Dion.