# Breaks the product based on ${x_j}$?

Can anyone help me to break this product into the series based on ${x_j}$ ?

$$\prod_{i=1}^{K}(1-x_i)$$

I want to break it to some function as below:

$$\sum_j\Psi(x_j)$$

I saw something like this long time ago in an article, but I cant remember it right now! I appreciate any suggestion for starting point! or any hint.

• I guess you mean $x^i$ in the product? – gammatester Jul 17 '15 at 12:09
• In your product there are $K$ numbers $x_1, \dots x_K$ but no $x$! – gammatester Jul 17 '15 at 12:14
• have you tried to derive a formula for $K=1,K=2$ etc. and find a pattern? – Surb Jul 17 '15 at 12:15
• @ Surb, yes it does not make sense ! maybe i made mistake – Cardinal Jul 17 '15 at 12:16
• Your edit will be hard to reach, since there are mixed terms appearing in your product. E.g. $K=2 \implies (1-x_1)(1-x_2)=1-x_1-x_2+{\color{red}{x_1x_2}}$ – Surb Jul 17 '15 at 12:21

• I'm not sure what you mean by break but I imagine it would be difficult to find a function $\Psi$ since asymptotically you could consider the Taylor expansions of $\log (1-x_{i})$ etc – Autolatry Jul 17 '15 at 12:41