1. ## Series theorem proof

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

2. ## Re: Series theorem proof

Originally Posted by Duke
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
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.

3. ## Re: Series theorem proof

I don't know what the $\displaystyle n$ you talk about is.
To show the result, notice that if $\displaystyle \sum_{k=0}^{+\infty} a_k$ is convergent then $\displaystyle \sum_{n=0}^{+\infty}\sum_{k=2^n}^{2^{n+1}-1}a_k$ is convergent (it works because the $\displaystyle a_k$ are non-negative).

4. ## Re: Series theorem proof

Thanks for that proof drexel

5. ## Re: Series theorem proof

I would still like to know how rudin did it. He does very slick proofs.