to find the radius of convergence for the Maclaurin series of \(f(x) = \frac{1}{\sqrt{1 - x^2}}\), you can use the following simplified approach:
1. Start with the Maclaurin series for \(\frac{1}{\sqrt{1 - x^2}}\), which is known to be:
\[f(x) = \sum_{n=0}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2}x^{2n}\]
2. To find the radius of convergence (\(R\)), use the ratio test:
\[
R = \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right|
\]
where \(a_n\) is the coefficient of \(x^{2n}\) in the series.
3. Calculate \(a_n\) and \(a_{n+1}\) using the formula for \(f(x)\) above.
4. Take the limit as \(n\) approaches infinity to find \(R\).
This approach will give you the radius of convergence for the Maclaurin series.