# Determine the convergence of following series

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

• $a_n$ does not tend to $0$ so the series does not converge. Jun 27 at 23:15
• No test is required except what I mentioned in above comment. Always look at $\lim a_n$ before applying any test for convergence. Jun 27 at 23:18
If $$n$$ is even,$$a_n<\left(\frac{2+3n}{4+6n}\right)^n=\frac1{2^n},$$and therefore$$\frac{a_{n+1}}{a_n}>5\times2^n.$$So, by the ratio test, the series diverges.
But this is a complete waste of time. Since you don't have $$\lim_{n\to\infty}a_n=0$$, the series diverges. That's all.