Does this product sequence converge?

Posted by Jason Polak on 23. May 2018 · Write a comment · Categories: elementary · Tags: ,

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 »