IA idea · Pure maths, number & proof
Why does 1 + 1/4 + 1/9 + … equal π²/6?
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?
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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
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.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Not a natural fit | The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics. |
| AA HL | Good fit | Infinite series and convergence; Comparison with integrals to bound the tail |
| AI SL | Not a natural fit | The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach. |
| AI HL | Not a natural fit | The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level. |
Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: presenting Euler's argument as a complete proof; and error estimates without justification. Say how each one affects your answer.
Use of mathematics (E)
HL: infinite series and convergence; comparison with integrals to bound the tail; maclaurin series of sin x; factorising a polynomial by its roots; comparing coefficients, each explained in your own words and carried out correctly.
How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Use the same method to derive Σ1/n⁴ = π⁴/90.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.
See a complete IA, marked
How long does a skydiver take to reach terminal velocity? (AA HL) is a complete pure maths exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.
What mathematics does this IA use?
Infinite series and convergence, comparison with integrals to bound the tail, maclaurin series of sin x and factorising a polynomial by its roots; comparing coefficients.
Where can I get data for this IA?
No data needed. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
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Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · annotated exemplars, marked on every criterion
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