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? More »