2011-50.md

course	course_year	question_number	tags	title	year
Methods	IB	50	IB 2011 Methods	Paper 1, Section II, A	2011

Let $f(t)$ be a real function defined on an interval $(-T, T)$ with Fourier series

$$f(t)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n} \cos \frac{n \pi t}{T}+b_{n} \sin \frac{n \pi t}{T}\right)$$

State and prove Parseval's theorem for $f(t)$ and its Fourier series. Write down the formulae for $a_{0}, a_{n}$ and $b_{n}$ in terms of $f(t), \cos \frac{n \pi t}{T}$ and $\sin \frac{n \pi t}{T}$.

Find the Fourier series of the square wave function defined on $(-\pi, \pi)$ by

$$g(t)=\left{\begin{array}{lr} 0 & -\pi<t \leqslant 0 \\ 1 & 0<t<\pi \end{array}\right.$$

Hence evaluate

$$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k+1)}$$

Using some of the above results evaluate

$$\sum_{k=0}^{\infty} \frac{1}{(2 k+1)^{2}}$$

What is the sum of the Fourier series for $g(t)$ at $t=0$ ? Comment on your answer.