Partial Differential Equations: Exercises

08 Convergence of Fourier series, part 2

Exercise 1: Summing infinite series using Fourier series

Recall the method used in Exercise 1 and 2 in class of using a Fourier series to sum an infinite series. This exercise is a continuation of that method, and it makes use of the table of Fourier series that we’ve used in previous exercises.

For each of the following series, use the table linked above to find the sum of the series.

  1. \(\displaystyle\sum_{n=1}^\infty \frac{1}{4n^2-1}\)

  2. \(\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}\)

  3. \(\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1)^3}\)

  4. \(\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2}\)

  1. \(\frac{1}{2}\)

  2. \(\frac{\pi^2}{6}\)

  3. \(\frac{\pi^3}{32}\)

  4. \(\frac{\pi^2}{12}\)