Calc6.23

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.