## Archive for April 30th, 2012

### A look at a few Tripos questions III

April 30, 2012

Here’s another one.

10F. State without proof the Integral Comparison Test for the convergence of a series $\sum_{n=1}^\infty a_n$ of non-negative terms.

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

In each of the following cases determine whether the series is convergent or divergent:

(i) $\displaystyle \sum_{n=3}^\infty \frac 1{n\log n}$,

(ii) $\displaystyle \sum_{n=3}^\infty \frac 1{n\log n(\log\log n)^2}$,

(iii) $\displaystyle \sum_{n=3}^\infty \frac 1{n^{1+1/n}\log n}$.
(more…)