Solution
= Solution
Factor out $9$ and use the <binomial series> with exponent $-1/2$:
$$
(x^2+9)^{-1/2}=\frac13\left(1+\frac{x^2}{9}\right)^{-1/2}
=\frac13\left[1-\frac12\frac{x^2}{9}+O(x^4)\right].
$$
Thus the expansion through cubic order is
$$
\boxed{\frac1{\sqrt{x^2+9}}=\frac13-\frac{x^2}{54}+O(x^4).}
$$
The <coefficients> of $x$ and $x^3$ vanish because the <function> is an <even function>. The underlying binomial expansion converges for $|x|<3$.