Derive power series for $\frac{1}{1-x^2}$ from $\frac{1}{1-x}$? The series $\sum_{k=0}^{\infty} x^k$ is known and equals $\frac{1}{1-x}$ for $|x|<1$.
Can I use this fact and derive the power series for $\frac{1}{1-x^2}$ from it, eg using $\frac{1}{1-x^2}=\frac{1}{1-x}\frac{1}{1+x}, x\neq\pm1$. Or must I derive it from the definition $\sum_{k=0}^{\infty} \frac{f^k(0)}{k!}x^k$
Thanks in advance!
Alexander

First way:
$$\frac1{1-x^2}=\frac1{1-(x^2)}=1+(x^2)+(x^2)^2+\ldots=1+x^2+x^4+\ldots$$
Second way: using that
$$\frac1{1+x}=\frac1{1-(-x)}=1+(-x)+(-x)^2+(-x)^3+\ldots=1-x+x^2-x^3+\ldots$$
we decompose by partial fractions
$$\frac1{1-x^2}=\frac12\left(\frac1{1-x}+\frac1{1+x}\right)=
\frac12\left(1+x+x^2+\ldots+1-x+x^2-x^3+\ldots\right)=$$
$$=\frac12\left(2+2x^2+2x^4+\ldots\right)=1+x^2+x^4+\ldots$$
