Analysis and calculus · Maths EE idea · Solid
How fast do series for π converge?
A research question to start from
How quickly do the Leibniz series and Machin-type formulas converge to π, and how can the error after n terms be bounded?
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
A clear measurable outcome (error bounds) with proofs and computation that check each other.
Mathematics you would need
- Alternating series and error bounds
- Taylor series of arctan x
- Trigonometric identities for Machin formulas
- Rates of convergence
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
- Derive the arctan series and the Leibniz series.
- Prove the alternating-series error bound and test it.
- Prove a Machin-type identity and compare convergence rates.
- Discuss a convergence-acceleration technique.
Scope and difficulty
Solid. Solid; restrict to two or three series.
Pitfalls
- Racing through many formulas.
- No proofs of the identities.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Machin formula proof arctan; alternating series estimation theorem. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).