Determine the convergence of following series. $$ \sum_{n=1}^\infty a_n$$ where $$ a_n = \begin{cases} \bigg( \dfrac{2+3n}{5+6n} \bigg)^n, & \text{if $n$ is even} \\ 5 & \text{if $n$ is odd} \end{cases} $$
As I think this series obviously divergence series because the when $n$ is odd $a_n=5$, so all the partial sum of odd terms divergence
But how can I prove it using Root test or Ratio test?
I used root test then I got
$L=\lim \sup |5^\frac{1}{n}|=1$ so test is inconclusive
Can anyone give some idea about this problem?
Thank you!