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[parent] a series related to harmonic series (Example)

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 term 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.




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See Also: $p$ test, Raabe's criteria

Keywords:  series with positive terms

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Cross-references: implies, comparison test, diverges, harmonic series, integer, positive, divergent, series

This is version 2 of a series related to harmonic series, born on 2008-03-25, modified 2008-03-25.
Object id is 10441, canonical name is ASeriesRelatedToHarmonicSeries.
Accessed 579 times total.

Classification:
AMS MSC40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

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