How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series? How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series? I know how to expand $\frac{1}{\sqrt{1-x}}$, but I have no idea how to expand $\frac{1}{\sqrt{1-x^2}}$. Simply differentiate this makes expanding way to complicated.

If $$f(x) = \sum_{k=0}^\infty a_k x^k,$$
then $$f(x^2) = \sum_{k=0}^\infty a_k x^{2k}.$$
This means that if you define $$b_k=\begin{cases}a_{l}& k=2l\\0&\text{otherwise}\end{cases},$$
you have $$f(x^2) = \sum_{k=0}^\infty b_k x^k$$
