Convergence/divergence of a series: $a_1 = 2$, $a_{n+1}=\frac{5n + 1}{4n+3}a_n$

Question

Given the series

The terms of a series are defined recursively by the equations $$a_1 = 2\\ a_{n+1}=\frac{5n + 1}{4n+3}a_n$$

Determine whether the summation of $a_n$ converges or diverges.

Answer 1

Notice that for $n > 2$, $\frac{a_{n+1}}{a_n} > 1 \implies a_{n+1} > a_n > 0$.

Thus, the summation (I'm assuming from $n=1$ to $n = \infty$) of this strictly increasing (for $n > 2$) sequence is divergent.

Answer 2

For converging series $$\lim_{n \rightarrow \infty} \frac {a_{n+1}}{a_{n}} <1$$ by Ratio Test

but here $$\lim_{n \rightarrow \infty} \frac {a_{n+1}}{a_{n}} =\frac{5n + 1}{4n+3}=5/4>1 $$
so It diverges