# a series related to harmonic series

The series

 $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n\sqrt[n]{n}}=\sum_{n=1}^{\infty}% \frac{1}{n^{1+\frac{1}{n}}}$ (1)

is divergent.  In fact, since for every positive integer n, one has  $2^{n}>n$,  i.e.  $\sqrt[n]{n}<2$, any of the series satisfies

 $\frac{1}{n\sqrt[n]{n}}>\frac{1}{2n}.$

Because the harmonic series and therefore also $\sum_{1}^{\infty}\frac{1}{2n}$ diverges, the comparison test implies that the series (1) diverges.

Title a series related to harmonic series ASeriesRelatedToHarmonicSeries 2013-03-22 17:56:40 2013-03-22 17:56:40 pahio (2872) pahio (2872) 5 pahio (2872) Example msc 40A05 PTest RaabesCriteria