Calc6.72

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}\left({\frac  {15n-6}{4n+2}}\right)^{{7n}}.

Using the root test, we have

\lim _{{n\rightarrow \infty }}{\sqrt[ {n}]{\left({\frac  {15n-6}{4n+2}}\right)^{{7n}}}}=\lim _{{n\rightarrow \infty }}\left({\frac  {15n-6}{4n+2}}\right)^{7}=\left({\frac  {15}{4}}\right)^{7}>1

So, by the root test, this series diverges.


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