I tried simplifying the product $$\prod_{k=1}^{\infty}\left[1-x^k\right]$$ by factoring it into $$\prod_{k=1}^{\infty}\left[\left(1-x\right)\sum_{i=0}^{k-1}x^i\right].$$ I am not very experienced in solving complicated products , and couldn't seem to find any solving method on the internet. Is there any way of converting the above expression into a mere series and taking the limit, or is there a better way of approaching this problem?
EDIT: By solving I mean finding an expression for $x$ which is not in terms of a product.
This product is related to partitions. I let Mathematica compute the first terms by actually doing the product. The result is $$\eqalign{f(x)&=1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26}\cr &\quad - x^{35} - x^{40} + x^{51} + x^{57} - x^{70} - x^{77} + x^{92} + x^{100}-\ldots\quad.\cr}$$ One quickly realizes that there is a law in the exponents. It leads to $$f(x)=1+\sum_{n=1}^\infty (-1)^n(1+x^n) x^{(3n^2-n)/2}\ .$$ (This is not a proof. It just shows what hand computing would furnish in the end.)