Calc6.22

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n{\sqrt  {n^{2}+1}}}}.

The square root of n^{2}+1 acts a lot like n so this function acts a lot like {\frac  {1}{n^{2}}}. Let us use the direct comparison test to see if our series converges or diverges.

{\frac  {1}{n^{2}}}>{\frac  {1}{n{\sqrt  {n^{2}+1}}}}\,

Since the terms are less, term by term, than a series which is known to converge (because it is a p-series with p=2) by the direct comparison test, this series converges.


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