Re: The series converges?

Hey xinglongdada.

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

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?

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.

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...