Oh yeah, Rudin gives a bad proof for that, it's a little sloppy. This theorem is known as Cauchy's Condensation test, a proof can be found here.
This is a proof from Principles of mathematical analysis. A series of non negative decreasing terms a_k converges iff the series 2^k*a_2^k converges. I need to show their sequence of partial sums are either both bounded or both unbounded. What I don't understand is he lets n<2^k but does he mean this is for k=n or something else?
Thanks