Exercise 5.8: Fourier analysis

1. Consider the following series $f(t)=\frac{2}{\pi }\sum_{n=1}^{\infty }\sin {nt}.$Plot $f(t)$ retaining only the first 3 terms. Increase the number of terms until you are satisfied that you are converging to $f(t)$ with some arbitrary but sufficient accuracy. What is the function represented by the sum?

   Figure 10: The ``sawtooth'' function studied in Exercise 5.8

   

2. Use the analytical expression for the Fourier coefficients and calculate the integrals using the ``sawtooth'' function depicted in Fig 5.2 to show that they are effectively given by $a_{n}=0$ and $b_{n}=(1/n\pi )(-1)^{n-1}$

3. What function os represented by the Fourier series with coefficients $% a_{0}=0$ and $a_{n}=b_{n}=1/n^{2}$ for $n\ne 0$?