Let $f(x) = \sqrt{1 - x^2}$ for $x \in [-1, 1]$.
(a) (7 marks) Using Binomial Theorem, prove that for $x \in (-1, 1)$, $f(x) = \sum_{n=0}^{\infty} (-1)^n c_n x^{2n}$, where
$c_n = \frac{(-1)^{n-1} (2n)!}{2^{2n} (n!)^2 (2n - 1)}$
for $n = 0, 1, 2, \dots$
(b) (9 marks) Using the fact that $\lim_{k \to \infty} \frac{k!}{\sqrt{2\pi k} (\frac{k}{e})^k} = 1$, show that $\sum_{n=0}^{\infty} c_n$ converges absolutely.
(c) (9 marks) Using (a) and (b), or otherwise, prove that
$\frac{\pi}{4} = 1 - \sum_{n=1}^{\infty} \frac{(2n)!}{2^{2n} (n!)^2 (2n - 1)(2n + 1)}$
(Hint: consider $\int_0^1 f(x) dx$.)
