Take as a given that is an increasing sequence with . (It is true: see here.)
For we know that
but that same sum diverges for any other r.
Let . Then, by definition of limit, there exists such that for each , . Thus for any , there are an infinite number of terms such that . Hence, is greater than any (monic) convergent geometric series, and is therefore divergent.
Anyone? I'm afraid that this post has had so many replies that nobody wants to read it, but I haven't got a usable answer. Plato gave a good hint maybe, but I don't follow it..