Prove that ?(an-an+1) converges if an only if limn->?an exists and is finite. Moreover, prove that if limn->?an=L, then ?(an-an+1) = a0 - L.



(both of the series go from n=0 to infinity, also, if you can't see very well, both of the limits are as n approaches infinity)

My attempt:

To prove this, as with any other biconditional, I start by assumng the seris an-an+1 converges and try to prove that the limit exists and is finite. So, define the series as a sum of its partial sums. let s1=a1, s2=a1+a2 and so on. Then when you add sum everything, you end up with just -an+1. And because the series converges, then -an+1 has to be finite, so the limit has to exist and be finite? This isn't enough, but I'm not sure what else to say. As for going the other way (assuming the limit exists and is finite) I have no idea what to do..



Also, this looks like an alternating series. Is there some conclusion of the alternating series test that makes all this much more simple?