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

Since we have a function to the nth power, the root test will probably be a good test to use here.

\lim _{{n\rightarrow \infty }}{\sqrt[ {n}]{\left({\frac  {n}{2n+1}}\right)^{n}}}=\lim _{{n\rightarrow \infty }}{\frac  {n}{2n+1}}={\frac  {1}{2}}<1

Since the limit is less than 1, this series converges by the root test.

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