Chapter 11.2, Problem 63E

To determine

(a) How to show that

$\frac{1}{1+x^2}=1-x^2+x^4-\mathrm{...}+{\left(-1\right)}^{N-1}{x}^{2N-2}+\frac{{\left(-1\right)}^N{x}^{2N}}{1+x^2}$

(b) How to show by integrating over $\left[0,1\right]$

$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{...}+\frac{{\left(-1\right)}^{N-1}}{2N-1}+{\left(-1\right)}^N{\displaystyle \underset{0}{\overset{1}{\int }}\frac{x^{2N}dx}{1+x^2}}$

(c) How to prove that

$0\le {\displaystyle \underset{0}{\overset{1}{\int }}\frac{x^{2N}dx}{1+x^2}\le \frac{1}{2N+1}}$

(d) How to prove that

$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{...}$