Continuous Compounding?

Question

How does A=P(1+r/n)^nt become A = Pe^(rt)? Can someone show me a mathematical formula?

Answer 1

EDIT: Left_Field is right, n = no of periods. Corrections below


It is done by calculus. Continuous compounding is

A = lim(n->∞) P*(1+r/n)^nt

= lim(n->∞) P*[(1+r/n)^n]^t

lim(n->∞) (1+r/n)^n = e^r, e is the base of the natural logs

A = P*[e^r]t = P*e^rt

Answer 2

Start by writing out (1+r/n)^nt long hand with n= some smallish number like 3: (1+r/n)(1+r/n)(1+r/n). Then do it again with 4 then 5 terms. Hopefully, you will see a pattern emerge as you multiply the terms so you can write the equation for any value for "n". I think you will find that the equation starts to look like the infinite series representation for the exponential function (see link). As your compounding goes continuous, "n" approaches infinity - thus the infinite series.

p.s. gp4rts' solution is more elegant, except that n approaches infinity instead of 0