Calc6.71

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\left({\frac  {7n}{12n-6}}\right)^{{2n}}.

\lim _{{n\rightarrow \infty }}{\sqrt[ {n}]{\left({\frac  {7n}{12n-6}}\right)^{{2n}}}}=\lim _{{n\rightarrow \infty }}\left({\frac  {7n}{12n-6}}\right)^{2}=\lim _{{n\rightarrow \infty }}{\frac  {49n^{2}}{144n^{2}-144n+36}}={\frac  {49}{144}}<1

Thus, since this limit is less than 1, the series converges absolutely by the ratio test.


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