jaydfox Wrote:Well, most of real and complex analysis would fall apart if limiting cases were not sufficient to provide proofs!

If you ever looked into an analysis book you would know that there are quite rigorous proofs for convergence. You merely telled something about an that goes rapidly to 0. Neither is clear whether this is inside or outside the parenthesis nor why this would imply the convergence of the sequence. And about what limiting cases do you speak?

A proof could perhaps look like this:

We want to show that the sequence converges.

By the law we inductively construct a supplemental sequence by

and .

This sequence is exactly chosen such that

particularly

.

Now it is clear by looking at the derivative of that for each . If we repeatedly apply this to the formula of , while assuming that and hence , we get

and further for

Now is , because for . But we know that the series converges for and hence is the sequence bounded from above.

An induction shows that is increasing.

We show (for arbitrary ) by induction over m.

Induction base:

and for the induction step show it for :

From the assumption follows by monotone increase of :

then

which yields , the induction assertion.

So particularely is increasing and bounded from above (for ) and so has a limit, given that and that .

Theorem. The sequence converges if , (and ).