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For example, lets look at $\sin\pi z = \prod\limits_{n\in\mathbb{Z}-\{0\}}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$.

For all n, $k_n = 2$. With this $k_n$, the product is entire. What is $k_n$? I know for that product we can write it as $\prod\limits_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$. So $k_n$ is the power of z and n. I don't know why this was chosen.

Then there is the corollary:

If $\sum\limits_{n =1}^{\infty}\frac{1}{\left|n\right|^2}$ converges, then $\prod\limits_{n=1}^{\infty}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$ converges.

I guessing this all related to picking $k_n$ but I don't get it.