Analysis | Part IA, 2003

State without proof the Integral Comparison Test for the convergence of a series n=1an\sum_{n=1}^{\infty} a_{n} of non-negative terms.

Determine for which positive real numbers α\alpha the series n=1nα\sum_{n=1}^{\infty} n^{-\alpha} converges.

In each of the following cases determine whether the series is convergent or divergent: (i) n=31nlogn\sum_{n=3}^{\infty} \frac{1}{n \log n}, (ii) n=31(nlogn)(loglogn)2\sum_{n=3}^{\infty} \frac{1}{(n \log n)(\log \log n)^{2}}, (iii) n=31n(1+1/n)logn\sum_{n=3}^{\infty} \frac{1}{n^{(1+1 / n) \log n}}.

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