Consider again the function in Example 1 and its Fourier series (20).
f(x)=1-{\frac{8}{\pi^{2}}}\Biggl(\cos\biggl({\frac{\pi x}{2}}\biggr)+{\frac{1}{3^{2}}}\cos\biggl({\frac{3\pi x}{2}}\biggr)+{\frac{1}{5^{2}}}\cos\biggl({\frac{5\pi x}{2}}\biggr)+\cdots\Biggr)\\=1-\frac{8}{\pi^{2}}\sum_{m=1,3,5,…}^{\infty}\frac{1}{m^{2}}\cos\left(\frac{m\pi x}{2}\right)\\=1-\frac{8}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\cos\biggl(\frac{(2n-1)\pi x}{2}\biggr). (20)
Investigate the speed with which the series converges. In particular, determine how many terms are needed so that the error is no greater than 0.01 for all x.