Can we say $(\prod_{n=1}^{\infty}n)^2=(\prod_{n=1}^{\infty}n^2)$

Question

Can we say $$(\prod_{n=1}^{\infty}n)^2=(\prod_{n=1}^{\infty}n^2)$$

It works fine if things are finite, does it hold in $n$ goes to infinity?

Answer 1

You can do that (but you probably mean $x_n$ instead of $n$ in the product). A bit more formal argument goes as follows: For any sequence $(x_n)_{n\in\mathbb N}$ of real numbers, define $$\prod_{n=1}^{\infty} x_n\equiv \lim_{N\to\infty}\left(\prod_{n=1}^N x_n\right)$$ whenever the limit on the right-hand side exists. Now suppose that this limit does exists. Then, since the function $x\mapsto x^2$ is continuous, one has that $$\left(\prod_{n=1}^{\infty} x_n\right)^2=\left\{\lim_{N\to\infty}\left(\prod_{n=1}^N x_n\right)\right\}^2=\lim_{N\to\infty}\left\{\left(\prod_{n=1}^N x_n\right)^2\right\}=\lim_{N\to\infty}\left(\prod_{n=1}^N x_n^2\right)=\prod_{n=1}^{\infty} x_n^2.$$ The penultimate equality follows from the analogous formula for finite products.