Use the limit comparison test to determine whether $\sum_{n=1}^{\infty} \frac{7 n^{5}}{n^{6}+n^{2}+2}$ converges.
A)The series diverges.
B)The series converges.

Question

Quizplus · Accepted Answer

The limit comparison test involves comparing the given series with a simpler series. Here, compare $\frac{7n^5}{n^6 + n^2 + 2}$ with $\frac{1}{n}$. The dominant term in the denominator is $n^6$, so the series behaves like $\frac{7n^5}{n^6} = \frac{7}{n}$, which is a harmonic series and diverges. Thus, the original series also diverges.

Use the Limit Comparison Test to Determine Whether ∑n=1∞7n5n6+n2+2\sum_{n=1}^{\infty} \frac{7 n^{5}}{n^{6}+n^{2}+2}∑n=1∞​n6+n2+27n5​

Use the Limit Comparison Test to Determine Whether $\sum_{n=1}^{\infty} \frac{7 n^{5}}{n^{6}+n^{2}+2}$