Consider Gregory's expansion

\[\tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\cdots=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2 k+1} x^{2 k+1}\]

a. Derive Gregory's expansion using the definition

\[\tan ^{-1} x=\int_{0}^{x} \frac{d t}{1+t^{2}}\]

expanding the integrand in a Maclaurin series, and integrating the resulting series term by term.

b. From this result, derive Gregory's series for \(\pi\) by inserting an appropriate value for \(x\) in the series expansion for \(\tan ^{-1} x\).