$\prod_{n=1}^\infty \frac{2n+1}{2n}$ diverges This is the Problem from the textbook "Intro to analysis" by Arthur Mattuck

Prove that $a_k := \prod_{n=1}^k \frac{2n+1}{2n}$ is strictly increasing and not bounded above.

Strict increasing is trivial: $\frac{a_{k+1}}{a_k} > 1$. But I am stuck at showing $a_k$ is not bounded above. If it is bounded above, then it must have a limit, so I think it suffices to show it diverges. But I also don't know how to show it's divergent.
Could anyone give me some hint on this?

$(a_n)$ is not bounded by above if and only the sequence defined by if$$\log(a_n)=\sum_{k=1}^n\log\left(1+\frac{1}{2k}\right),$$
is not bounded by above. Since $$\log\left(1+\frac{1}{2n}\right)\sim \frac{1}{2n},$$
this should be rather clear.
