Analysis and calculus · Maths EE idea · Ambitious
Three proofs that Σ1/n² = π²/6
A research question to start from
How do Euler's product argument, a double-integral proof and a Fourier-series proof each establish that the sum of 1/n² is π²/6, and which is most rigorous?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Comparing proofs gives you a natural line of argument and a strong evaluation section.
Mathematics you would need
- Convergence of series
- Taylor series of sin x
- Double integrals (or Fourier series)
- Rigour and limiting processes
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Present Euler's argument and identify the steps that need justification.
- Work through one rigorous proof in full.
- Compare what each proof assumes and explains.
Scope and difficulty
Ambitious. Ambitious; two proofs done well beat three done briefly.
Pitfalls
- Copying proofs without commentary.
- Ignoring the gaps in Euler's original argument.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Basel problem proofs comparison; Euler sine product. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).