Analysis and calculus · Maths EE idea · Solid
Why do all the definitions of e agree?
A research question to start from
How can the limit (1 + 1/n)^n, the series Σ1/n! and the condition d/dx(eˣ) = eˣ be shown to define the same number?
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 question about rigour itself; you prove equivalences rather than assume them.
Mathematics you would need
- Limits of sequences
- The binomial theorem
- Series and convergence
- Derivatives of exponentials
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
- Show the limit exists (monotone and bounded).
- Prove the limit equals the series.
- Link both to the derivative property.
- Compare rates of convergence.
Scope and difficulty
Solid. Solid; careful limit arguments are the challenge.
Pitfalls
- Circular arguments.
- Numerical evidence used as proof.
Where to start reading
Search a library catalogue or a university's open lecture notes for: equivalence definitions of e proof; monotone convergence (1+1/n)^n. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).