Test the series below for convergence using the Ratio Test.

$\displaystyle {\sum_{{{n}={0}}}^{\infty}}\frac{{{\left(-{1}\right)}^{{n}}{8}^{{{2}{n}+{1}}}}}{{{\left({2}{n}+{1}\right)}!}}$

The limit of the ratio 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 converges diverges fails the ratio test .