Does the Series \(\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}\) Converge?

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The series \(\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}\) does not converge. As \(n\) approaches infinity, \(\sqrt[n]{n}\) approaches 1, making the terms of the series behave similarly to those of the harmonic series. The difference between the partial sums of this series and the harmonic series approaches a constant as \(k\) increases. Since the harmonic series is known to diverge, this series also diverges. Therefore, the conclusion is that the series diverges.
steviet
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\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}\]

Does this series converge?
 
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No it doesn't, because \sqrt[n]{n}} approaches 1 as n approaches infinity. Hence if we were to the the first k partial sums of the series in your post, and the harmonic series, the difference would approach a constant as k approached infinity. Since the harmonic series diverges, so does this one.
 

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