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…)


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