The series converges?

Printable View

• Nov 9th 2012, 07:46 PM
xinglongdada
The series converges?
Let $a_n$ be positive, and $\sum_{n=1}^\infty a_n$ be divergent. Assume $S_n=a_1+\cdots+a_n$. Prove that for any $\sigma>1,$ the series $\sum_{n=1}^\infty \frac{a_n}{S_n^\sigma}$ converges.
• Nov 10th 2012, 01:56 AM
chiro
Re: The series converges?
Hey xinglongdada.

Consider that a/n + b/n = (a+b)/n and then consider the ratio test.
• Nov 10th 2012, 02:45 AM
xinglongdada
Re: The series converges?
Do you mean by considering $\sum_{n=1}^\infty (S_n-S_{n-1})/S_n^\sigma$, then using ratio criterion? I do not know how to do. Would you be more precise?
• Nov 10th 2012, 03:36 AM
chiro
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.
• Nov 10th 2012, 05:10 AM
xinglongdada
Re: The series converges?
In fact, we need have $|a_{n+1}/a_n| for some fixed $r$ to guarantee the convergence of the series. So I do not know how to take the limit...