Exposition: Suppose you lent $1 at a 100% loan for 10 years.
- Compound once you get (1+1)^1 = $2
- Compound twice you get (1+1/2)^2 = 2.25
- Compound thrice you get (1+1/3)^3 = 2.3737
- Compound 10 times you get 2.5937
What happens as you compound more? The kids' initial guesses were that you would get an unbounded return or that you would get $3 eventually. We saw that it seemed to zero in on 2.718 by doing a million times then a billion times (at which point the calculators started producing random values in low order digits). Then I asked them to find e^1 on their calculator.