Euler's number

See also exponential function.

Interest proof

A result that shows up often in discussion of compound interest is:

$$\underset{n\to \infty }{\lim }(1+\frac{r}{n}{)}^n=e^r$$

This turns out to be a fairly classical proof. Bernoulli proved it back in the 1700s, though there are many proofs out there.

We start by using the binomial expansion to get a feel for what we’re working with:

$$(1+\frac{r}{n}{)}^n=1^n+C_1^n{1}^{n-r}\frac{r}{n}+C_2^n{1}^{n-2}(\frac{r}{n}{)}^2+\cdots$$

where $C_r^n$ is the combination:

$$C_r^n=\frac{n!}{r!(n-r)!}$$

Simplifying the above a little:

$$=1^n+1^{n-1}r+\frac{n!}{2!(n-2)!}{1}^{n-2}(\frac{r}{n}{)}^2+\frac{n!}{3!(n-3)!}{1}^{n-3}(\frac{r}{n}{)}^3+\cdots$$

Just isolating one of the terms to see if we can simplify things:

$$\frac{n!}{2!(n-2)!}(\frac{r}{n}{)}^2=\frac{1}{2!}\frac{r^2(n-1)}{n}$$

We can show with L’Hôpital’s rule that:

$$\underset{n\to \infty }{\lim }\frac{n-1}{n}{1}^{n-2}=1$$

This turns out this is true for all of the $n$ coefficients (which we can probably prove inductively). Nevertheless, with this result we simplify the big term:

$$=1+r+\frac{r^2}{2!}+\frac{r^3}{3!}+\cdots$$

We might recognise that the Taylor series expansion for $e^x$ is:

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$$

i.e.,

$$=e^r$$

as stated.