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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n+4}{(n+2)(n+1)}}.

Another comparison test we can use is the limit comparison test. To do this test, we compare two series by looking at the ratio of their terms as n\rightarrow \infty . If this term is a finite positive number, then both series either converge or diverge.

\lim _{{n\rightarrow \infty }}{\frac  {{\frac  {1}{n}}}{{\frac  {n+4}{(n+2)(n+1)}}}}=\lim _{{n\rightarrow \infty }}{\frac  {(n+2)(n+1)}{n(n+4)}}=1\,

Since this is a finite, positive number, then both \sum _{{n=1}}^{{\infty }}{\frac  {n+4}{(n+2)(n+1)}} and \sum _{{n=1}}^{{\infty }}{\frac  {1}{n}} converge or both diverge. We know that \sum _{{n=1}}^{{\infty }}{\frac  {1}{n}} diverges so our series also diverges.

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