# Thread: infinite product convergence

1. ## infinite product convergence

i'm having trouble trying to expand the following infinite product .

$\prod^{\infty}_{n=1}(1-\frac{x^n}{n})$

so , does it converge ? and what to ? i know it has something to do with elliptic functions ....but don't know where to start !

2. Originally Posted by mmzaj
i'm having trouble trying to expand the following infinite product .

$\prod^{\infty}_{n=1}(1-\frac{x^n}{n})$

so , does it converge ? and what to ? i know it has something to do with elliptic functions ....but don't know where to start !
Note that $\prod_{n=1}^{\infty}\left\{1-\frac{x^n}{n}\right\}=\exp\left(\ln\prod_{n=1}^{\i nfty}\left\{1-\frac{x^n}{n}\right\}\right)$ consider that we may rewrite $\ln\prod_{n=1}^{\infty}\left\{1-\frac{x^n}{n}\right\}$ as $\sum_{n=1}^{\infty}\ln\left(1-\frac{x^n}{n}\right)$. So obviously $\prod_{n=1}^{\infty}\left\{1-\frac{x^n}{n}\right\}=\exp\left(\sum_{n=1}^{\infty }\ln\left(1-\frac{x^n}{n}\right)\right)$ is only going to be finite if $\sum_{n=1}^{\infty}\ln\left(1-\frac{x^n}{n}\right)$ is finite. For the value think the infinite product for $\text{sinc}(x)=\frac{\sin(\pi x)}{\pi x}$

EDIT: Note that the last part is incorrect. I do not know where my head was this is not the since function