Where does e come from? Compounding more and more often

AA SLAA HL Solid Also in: Pure maths, Calculus

Research question

How does the value of an investment change as interest is compounded yearly, monthly, daily and continuously, and why does (1 + 1/n)ⁿ approach e?

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

Why it makes a good exploration

The number e first appeared in a question about interest. Exploring compounding frequency numerically and then proving the limit connects finance with some of the most important ideas in AA.

The mathematics you'll need

Compound interest with n periods
Sequences and limits
The binomial expansion to show convergence
HL: limits using logarithms and L'Hôpital's rule

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; optionally compare with a bank's stated AER and nominal rates.

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 numerically with increasing n.
Conjecture the limit.
Show convergence with the binomial expansion.
Link AER and nominal rates.
Reflect on why continuous compounding is used in models.

Pitfalls that cost marks

Only numerical tables without explanation.
Hand-waving the limit.
Confusing AER with nominal rate.

Showing personal engagement

Compare savings accounts you could open.
Investigate the historical story of Jacob Bernoulli's question.
Estimate how fast the sequence converges.

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

Taking it further

HL: prove that (1 + 1/n)ⁿ is increasing and bounded.

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).