Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} f(x)&=1+x+x^2+x^3+x^4+... \\ f(x)&=1+x(1+x+x^2+x^3+...) \\ f(x)&=1+x \cdot f(x) \\ f(x)−x \cdot f(x)&=1 \\ f(x)(1−x)&=1 \\ f(x)&= \frac{1}{(1−x)} \end{split} }
To calculate Failed to parse (syntax error): {\displaystyle \frac{1}{(1−x^2)}}
we just replace x with
Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} f(x^2)&=1+x^2+(x^2)^2+(x^2)^3+(x^2)^4+... \\ &=1+x^2+x^4+x^6+x^8... \\ \end{split} }