1. Find the Fourier sine series of f(x)=1, 0<x<L/2. Use this to prove that 1-1/3+1/5-1/7+...=Pi/4
2. Solve df/dt=d^2 f/dx^2 - f subject to the initial condition f(0,x)=1 if |x|<L/2 or f(0,x)=0 if |x|>L/2
Please see the attachment to view the questions with correct mathematical notation (and also phrased slightly differently)
Working with Fourier transforms and Fourier series is an extremely important technique in applied mathematics. This solution contains two questions; the first is a simple example of finding a Fourier series, then using this series to find the exact value of an infinite sum (a remarkable and beautiful application of Fourier series). The second shows how inverse Fourier transforms can be used to find solutions of PDEs. The solution is a 2 page Word document with equations written using Mathtype. Each step is explained.