One standard definition is e=sum_{n=0}^{infty} rac{1}{n!}. Each new factorial term is smaller than the last, so the sum converges quickly.

That makes this constant ideal for teaching approximation and convergence in a compact way.

Euler's number is the base of natural logarithms and shows up in growth, decay, compound interest, calculus, and probability.

Use this page to see how a finite series gets close to the constant before you move on to those applications.