Calc6.23

From Exampleproblems

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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Calc6.23

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