(Solved): This is a conceptual problem that will help us determine when the limit comparison test gives a de ...

This is a conceptual problem that will help us determine when the limit comparison test gives a desired result.. Find the value of \( p \) such that \( \lim _{n \rightarrow \infty} \frac{n^{P}}{\sqrt{7 n^{9}+2}} \) takes a positive finite value. Using the value of \( p \) you found above, evaluate the limit \( \lim _{n \rightarrow \infty} \frac{n^{p}}{\sqrt{7 n^{9}+2}}= \) With \( a_{n}=\frac{1}{\sqrt{7 n^{9}+2}} \) and \( b_{n}=\frac{1}{n^{\mathrm{p}}} \), using the value of \( p \) and the value of the limit above, since \( F \) 1, the series \( \sum_{n=1}^{\infty} b_{n}=\sum_{n=1}^{\infty} \frac{1}{n^{p}} \) We conclude that the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{7 n^{9}+2}} \)