Determine convergence or divergence of $a_n=\frac{4}{7}(1+\frac{1}{2n})^n$ $\sum_{n=1}^\infty a_n$ such that $a_1=1$, $a_n=\frac{4}{7}(1+\frac{1}{2n})^n$

I need to determine convergence or divergence of this series.

So I think the ratio test would work for this but i'm stuck when evaluating the limit.

$$\lim_{n\to \infty}\frac{(1+\frac{1}{2n+2})^{n+1}}{(1+\frac{1}{2n})^n}$$

Can anyone give me a hint as to how i should proceed?
The series $\sum_{n=1}^\infty a_n$ diverges since $a_n$ does not go to $0$ as $n\to\infty$. We have that
$$
a_n=\frac47\biggl(1+\frac1{2n}\biggr)^n=\frac47\sqrt{\biggl(1+\frac1{2n}\biggr)^{2n}}\to\frac{4\sqrt e}7
$$
as $n\to\infty$.
$$\lim _{x\rightarrow \infty} \left( 1+ \frac{1}{2x} \right) ^x \geq  \lim _{x\rightarrow \infty} 1^x = \lim _{x\rightarrow \infty} 1 = 1 \neq0 $$

since $a_n \not \rightarrow 0$ we have $\sum a_n = \infty$.