Calc6.39

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n}{3^{{n+1}}}}.

Let us use the root test to see if we can conclude anything.

\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {n+1}{3^{{n+2}}}}}{{\frac  {n}{3^{{n+1}}}}}}\right|=\lim _{{n\rightarrow \infty }}{\frac  {n+1}{3n}}={\frac  {1}{3}}\,

Since this is a value between 0 and 1, our series converges absolutely by the ratio test.


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