Under what conditions can we swap a limit with an infinite product? When is the following true: $$\lim_{n\to\infty}\prod^{\infty}_{k=1}a_{nk} = \prod^{\infty}_{k=1}\lim_{n\to\infty}a_{nk}$$
 A: You have
$$\lim_{n\to\infty}\prod^{\infty}_{k=1}a_{nk} = \prod^{\infty}_{k=1}\lim_{n\to\infty}a_{nk}$$
If and only if
$$\lim_{n\to\infty}\log(\prod^{\infty}_{k=1}a_{nk}) = \log(\prod^{\infty}_{k=1}\lim_{n\to\infty}a_{nk})$$ (Warning see footnote below 1)
Since $\log$ takes product to sum by setting $b_{nk}=\log(a_{nk})$ the above is equivalent to whether
$$\lim_{n\to\infty}\sum^{\infty}_{k=1}b_{nk} = \sum^{\infty}_{k=1}\lim_{n\to\infty}b_{nk}$$
Now you can use the Dominated convergence theorem which in this setting says:
If $|b_{nk}|\leq c_k$ for a sequence $c_k$ for which $\sum_{k=1}^\infty c_k$ exists and $\lim_{n\rightarrow\infty}b_{nk} = b_k$ then $\sum_{k=1}^\infty b_k$ exists and
$$\lim_{n\to\infty}\sum^{\infty}_{k=1}b_{nk} = \sum^{\infty}_{k=1}b_{k}$$
Translated back to products, you need that $|a_{nk}|\leq c_k$ for a sequence such that $\prod_{k=1}^\infty c_k$ exists then if $\lim_{n\rightarrow\infty} a_{nk} = a_k$ then $\prod_{k=1}^\infty a_k$ exists and
$$\lim_{n\to\infty}\prod^{\infty}_{k=1}a_{nk} = \prod^{\infty}_{k=1}a_{k}$$

1 This approach only works if the $a_{nk}$ are positive. Otherwise you can't take $\log$.
