How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series?

Question

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.

Answer 1

Simply substitute $x^2$ for $x$ in the expansion of$(1-x)^{-1/2}$

Answer 2

HINT:

$$\frac1{\sqrt{1-x^2}}=(1-x^2)^{-\frac12}$$

Answer 3

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 $$f(x^2) = \sum_{k=0}^\infty b_k x^k$$