## Does this product sequence converge?

Consider the following series: \begin{align*} a_1 &= \frac{4}{3}\\ a_2 &= \frac{4}{3}\frac{9}{8}\\ &\vdots\\ a_n &= \frac{4}{3}\frac{9}{8}\cdots\frac{(n+1)^2}{(n+1)^2-1} \end{align*} In other words, $a_n$ is the product of all the numbers of the form $n^2/(n^2 – 1)$ for $n=2,\dots, n+1$. Does $\lim_{n\to\infty} a_n$ exist?