Find the power series for a shifted center

Question

I know the power series of $f(x)=\frac{1}{1-x^2}$ is $\sum_{i=0}^\infty x^{2i}$. How would you get $f(x)$ expanded about, say, 10? Finding the nth derivative seems difficult since it's complicated by need for the quotient rule.

Answer 1

$$\frac {1}{1-x^2}= (\frac {1}{2})(\frac {1}{1-x}+\frac {1}{1+x})$$

$$\frac {1}{1-x} = \frac {1}{-9-(x-10)}=(\frac {-1}{9})(\frac {1}{1+(x-10)/9 })$$

And $$\frac {1}{1+x} = \frac {1}{11+(x-10)}=(\frac {1}{11})(\frac {1}{1+(x-10)/{11}})$$

Each one of these partial fractions easily expands as a power series about $x=10.$

You can take it from there.