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

To use the root test on this, it will be convenient to rewrite what we have using the rules for powers.

{\frac  {(n!)^{n}}{(n^{n})^{2}}}={\frac  {(n!)^{n}}{n^{{2n}}}}={\frac  {(n!)^{n}}{(n^{2})^{n}}}

Now, we can use the root test on this series easily.

\lim _{{n\rightarrow \infty }}{\sqrt[ {n}]{{\frac  {(n!)^{n}}{(n^{2})^{n}}}}}=\lim _{{n\rightarrow \infty }}{\frac  {n!}{n^{2}}}=\infty

Thus, since this limit goes to infinity, the series diverges by the root test.

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