Welcome to Three Solutions to the Basel Problem. In this Calculus II honors project, we will look at
- the convergence of the Basel series\[1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n^2}.\]
- differentiation under the integral sign.
- Fourier series and Parseval's identity.
- Euler's solution via Weierstrass factorization.
I believe that given your strong calculus backgrounds, solving the Basel problem using the above tools is within reach. But, you will have to think. Not all the covered material will be results or techniques you have previously seen, so some assignments may seem daunting. That being said, I will do my best to guide you through each solution at a comfortable pace.
Assignments and Deliverables
This honors project will be based on four problem sets. I strongly encourage you to work together on these, but working independently is fine. In either case, please submit your own work. We will meet semi-regularly to discuss your solutions and any questions.
You will also prepare slides and give a final talk on one of the three solutions.
Problem Sets
- HP02: Differentiation Under the Integral Sign
After confirming the hypotheses of differentiation under the integral sign, you will combine multiple integration techniques to eventually find, using power series, a solution (due to F. L. Freitas) to the Basel problem.
- HP01: Introduction to the Basel Problem
You will use various (familiar) convergence tests for series to get a feel for the Basel series. You will also approximate its sum.