1.II .9 F. 9 \mathrm{~F} \quad

Analysis I | Part IA, 2008

Investigate the convergence of the series (i) n=21np(logn)q\sum_{n=2}^{\infty} \frac{1}{n^{p}(\log n)^{q}} (ii) n=31n(loglogn)r\sum_{n=3}^{\infty} \frac{1}{n(\log \log n)^{r}}

for positive real values of p,qp, q and rr.

[You may assume that for any positive real value of α,logn<nα\alpha, \log n<n^{\alpha} for nn sufficiently large. You may assume standard tests for convergence, provided that they are clearly stated.]

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