Why does 1 + 1/4 + 1/9 + … equal π²/6?

AA HL Ambitious Also in: Calculus

Research question

How quickly do partial sums of Σ1/n² approach π²/6, and can Euler's argument using the Maclaurin series of sin x be made convincing at the level of AA HL?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

The Basel problem defeated mathematicians for decades until Euler's audacious solution in 1734. Numerically exploring convergence and then reconstructing Euler's argument — and its gaps — is a classic HL exploration.

The mathematics you'll need

Infinite series and convergence
Comparison with integrals to bound the tail
Maclaurin series of sin x
Factorising a polynomial by its roots; comparing coefficients

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explore partial sums numerically.
Bound the error with integrals.
Present Euler's argument step by step.
Identify the unjustified step (infinite products) honestly.
Reflect on rigour in mathematics.

Pitfalls that cost marks

Presenting Euler's argument as a complete proof.
Error estimates without justification.
Copying the argument without understanding it.

Showing personal engagement

Use the method to find Σ1/n⁴.
Accelerate convergence using your tail estimate.
Discuss how mathematicians later made the proof rigorous.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Use the same method to derive Σ1/n⁴ = π⁴/90.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).