Analysis and calculus · Maths EE idea · Ambitious
How good is Stirling's approximation?
A research question to start from
How can Stirling's approximation for n! be derived, and how does its relative error behave as n increases?
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 derivation plus an error analysis: the discussion writes itself if you measure the error carefully.
Mathematics you would need
- Logarithms and sums
- Integral approximations of sums
- The trapezium rule and error terms
- Limits
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
- Bound ln n! between integrals and derive a first approximation.
- Refine it to Stirling's formula (citing the constant's derivation if needed).
- Compute relative errors and fit their behaviour.
- Evaluate the role of correction terms.
Scope and difficulty
Ambitious. Ambitious; the √(2π) constant can be referenced with explanation.
Pitfalls
- Using the formula without deriving any part of it.
- Error analysis by eye.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Stirling approximation derivation integral; relative error Stirling. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).