$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 7 n ^ { 2 } + 1 } \text { using any }$ appropriate test.
A) converges
B) diverges

Question

$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { n } { 7 n ^ { 2 } + 1 } \text { using any }$ appropriate test. 
A) converges
B) diverges

Quizplus · Accepted Answer

We can use the limit comparison test with the series $\sum_{n=1}^\infty \frac{1}{n}$, which we know diverges since it's a $p$-series with $p=1$. We have $$\lim_{n	o\infty} \frac{\frac{n}{7n^2+1}}{\frac{1}{n}} = \lim_{n	o\infty} \frac{n^2}{7n^2+1} = \frac{1}{7}.$$ Since the limit is a positive finite number, and $\sum_{n=1}^\infty \frac{1}{n}$ diverges, we conclude that $\sum_{n=1}^\infty \frac{n}{7n^2+1}$ must also diverge. Therefore, the answer is (B).