First, multiplying the series by gives

or

To prove that the sequence converges we will prove two things

(i) the sequence is increasing and

(ii) it is bounded above

First off, I think it's clear that

(i) writing out the first few terms suggests its increasing. To prove this assume that

thus,

so

and by induction,

(ii) It is bounded by 2

The first few terms suggest this. Assume that

then

thus,

so again by induction, it is ture for all n. Since the sequence is increasing and bounded above, it must converge.