Question

10) Using Limit Comparison Test (LCT), the following series

Answer 1

Compare to the divergent series,

$\displaystyle\sum_{n=1}^\infty\frac1n$

Then by the limit comparison test, the given series also diverges, since the limit

$\displaystyle\lim_{n\to\infty}\frac{\frac{2n^5-1}{n^6+1}}{\frac1n} = \lim_{n\to\infty}\frac{2n^6-n}{n^6+1}=\lim_{n\to\infty}\frac{2-\frac1{n^5}}{1+\frac1{n^6}}=2$

is positive and finite.