Activity 8.6.3.
Our goal in this activity is to find a power series expansion for \(f(x) = \frac{1}{1+x^2}\) centered at \(x=0\text{.}\)
While we could use the methods of Section 8.5 and differentiate \(f(x) = \frac{1}{1+x^2}\) several times to look for patterns and find the Taylor series for \(f(x)\text{,}\) we seek an alternate approach because of how complicated the derivatives of \(f(x)\) quickly become.
(a)
What is the Taylor series expansion for \(g(x) = \frac{1}{1-x}\text{?}\) What is the interval of convergence of this series?
(b)
How is \(g(-x^2)\) related to \(f(x)\text{?}\) Explain, and hence substitute \(-x^2\) for \(x\) in the power series expansion for \(g(x)\text{.}\) Given the relationship between \(g(-x^2)\) and \(f(x)\text{,}\) how is the resulting series related to \(f(x)\text{?}\)
