Test the series below for convergence using the root test.

$\displaystyle {\sum_{{{n}={1}}}^{\infty}}{\left(\frac{{{3}{n}^{{2}}}}{{{8}{n}+{3}}}\right)}^{{n}}$

The limit of the root test simplifies to $\displaystyle \lim_{{{n}\rightarrow\infty}}{\left|{f{{\left({n}\right)}}}\right|}$ where

$\displaystyle {f{{\left({n}\right)}}}=$ Preview Question 1 Part 1 of 3   .

The limit is Preview Question 1 Part 2 of 3   .
(enter oo for infinity if needed)

Based on this, the series Select an answer fails the root test diverges converges .