Calc6.43

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

Our instinct should be that this series converges since the factorial function on the bottom increases much more quickly than the simple linear function on the top. The terms should go to 0 extremely quickly. However, to prove this, we must use the ratio test.

\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {n+1}{(n+3)!}}}{{\frac  {n}{(n+2)!}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac  {n+1}{n(n+3)}}\right|=0\,

Thus, since this goes to 0, by the ratio test, this series does indeed converge and converge absolutely.


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