# How would you prove this little inequality?

1. Mar 2, 2004

### stunner5000pt

i'm stuck trying to prove this little inequality:

(1+ 1/n)^n <= e <= (1+1/n)^n+1
is there a way to prove this without without resorting to power series for e (because we're not allowed to, and we don't know this yet) and also note that n is a natural number, (positive integer).

2. Mar 2, 2004

Here's a hint:

$$\lim_{n \to \infty} \Big( 1 + \frac{1}{n} \Big)^n = e$$

3. Mar 2, 2004

### stunner5000pt

ummmm

ok i know tha already i just dont know how to prove it give me hint on hwo to prove it

4. Mar 2, 2004

Just how formally do you want to prove it?

It's pretty easy to notice that for $n<\infty$, the left side is less than e. When $n = \infty$, it is exactly equal to e.

The same holds true for the right side, except that it's always greater than e except when $n = \infty$.

5. Mar 3, 2004

### matt grime

You do it by power series for (1+x)^n valid when |x|<1 (ie x=1/n)

6. Mar 3, 2004

### Wooh

Why can you not just reason that (1+1/n)^n has to be less than (1+1/n)^n * (1+1/n), as, until n => infinity, 1+1/n will always be a positive value above * a positive value that will always make the right hand larger.