[chrchang523]

I found it useful to walk through evaluation of a few elementary instances of this class using simpler methods, to put the main result in perspective. Specifically, replace the initial 3 exponent with 0 or 1.

If the exponent is 0, then you have the sum 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..., from Zeno's most famous paradox (https://en.wikipedia.org/wiki/Zeno%27s_paradoxes ). If you are fortunate, you previously learned that this converges to 1, and played around with this enough in your head to have a solid understanding of why. If you are less fortunate, I recommend pausing to digest this result.

Then, if the exponent is 1, you have the sum 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + ... .

What happens if we subtract (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...) from it? We have (1/4 + 2/8 + 3/16 + 4/32 + ...) left over.

Then, if we subtract (1/4 + 1/8 + 1/16 + 1/32 + ...) from the latter, we still have (1/8 + 2/16 + 3/32 + ...) left over.

Then, if we subtract (1/8 + 1/16 + 1/32 + ...) from the latter, we still have (1/16 + 2/32 + ...) left over.

Continuing in this fashion, we end up subtracting off

(1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...) + (1/4 + 1/8 + 1/16 + 1/32 + ...) + (1/8 + 1/16 + 1/32 + ...) + (1/16 + 1/32 + ...) + (1/32 + ...) + ...

and this converges to the main sum. And, from the exponent-0 result, we know this is just 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...