Monday, March 23, 2009

limit goes to e

We've done symbolic derivatives for polynomials with nonnegative exponents. We've also done a few simple limits. So this 5-minute math item is more advanced than some of the others Topic: What is the limit as n goes to infinity of (1+1/n)^n, proceeding by example, and using a calculator.

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.

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I'm research faculty at MIT, and Chief Architect at Tokutek.