1. ## The series converges?

Let $\displaystyle a_n$ be positive, and $\displaystyle \sum_{n=1}^\infty a_n$ be divergent. Assume $\displaystyle S_n=a_1+\cdots+a_n$. Prove that for any $\displaystyle \sigma>1,$ the series $\displaystyle \sum_{n=1}^\infty \frac{a_n}{S_n^\sigma}$ converges.

2. ## Re: The series converges?

Consider that a/n + b/n = (a+b)/n and then consider the ratio test.

3. ## Re: The series converges?

Do you mean by considering $\displaystyle \sum_{n=1}^\infty (S_n-S_{n-1})/S_n^\sigma$, then using ratio criterion? I do not know how to do. Would you be more precise?

4. ## Re: The series converges?

Basically the ratio criterion is that if |a_(n+1)/a_n| < 1 then the series converges where a_n is the n_th term of the series.

So basically you need to show that this happens given your series expansion.

5. ## Re: The series converges?

In fact, we need have $\displaystyle |a_{n+1}/a_n|<r<1$ for some fixed $\displaystyle r$ to guarantee the convergence of the series. So I do not know how to take the limit...