Let be positive, and be divergent. Assume . Prove that for any the series converges.

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- Nov 9th 2012, 07:46 PMxinglongdadaThe series converges?
Let be positive, and be divergent. Assume . Prove that for any the series converges.

- Nov 10th 2012, 01:56 AMchiroRe: 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 AMxinglongdadaRe: The series converges?
Do you mean by considering , then using ratio criterion? I do not know how to do. Would you be more precise?

- Nov 10th 2012, 03:36 AMchiroRe: 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 AMxinglongdadaRe: The series converges?
In fact, we need have for some fixed to guarantee the convergence of the series. So I do not know how to take the limit...