Let be positive, and be divergent. Assume . Prove that for any the series converges.
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Consider that a/n + b/n = (a+b)/n and then consider the ratio test.
Do you mean by considering , then using ratio criterion? I do not know how to do. Would you be more precise?
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.
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...
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