Calc6.42

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {3^{n}}{n^{2}+2}}.

Since this involves an exponential function, use the ratio test.

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

Since the limit is greater than 1, this series diverges by the ratio test. Of course, the terms go to \infty since an exponential function increases more quickly than a power function so we also know it diverges by the nth term test.


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