5.2.3

[STARTS AT 3:12]

Professor: So it all depends on that limit of the partial sums. So if you recognize that it's telescoping, we can go ahead and write the sequence of partial sums. This one is also telescoping. We're going to have terms cancel out. So this is 1 minus e to the 1 over k plus 1. 

Now as k goes to infinity, this exponent goes to 0, which means this approaches. And that means Sk approaches-- oh, that should be an e right there. Just plug in a 1, e to the one. So this approaches e minus 1, which means that our sum, 1 to infinity of e to the 1 over n minus e to the 1 over n plus 1 equals e minus 1.