# Proof of e = lim(1 + 1/n)^n

#### gerardhoyle

Hi I've been struggling with this proof;

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

I'll lay it out.

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

2: $$\displaystyle e^-1 = \frac{1}{e^1} = \lim_{n \to \infty} ( \frac{n }{n + 1})^n$$

3: Set x = n + 1 so that n = x - 1

4: $$\displaystyle e^-1 =\lim_{x \to \infty} (\frac{x - 1}{x})^x-1 =\lim_{x \to \infty} (\frac{x - 1}{x})^x(\frac{x - 1}{x})^1$$ (the latex messes up but that -1 is in the exponent in the first part here, as is obvious from the second part!).

5: $$\displaystyle = \lim_{x \to \infty} (1 - \frac{1}{x})^x(1 - \frac{ 1}{x})^1 = \lim_{x \to \infty} (1 - \frac{1}{x})^x \cdot 1$$

Now the proof tells me to return back to 'n' instead of x.

6: $$\displaystyle \lim_{x \to \infty} (1 - \frac{ 1}{x})^x \Rightarrow \ \lim_{n \to \infty} (1 - \frac{ 1}{n + 1})^n + 1$$ (where the +1 at the end here is in the exponent!)

7: $$\displaystyle \lim_{n \to \infty} (1 - \frac{ 1}{n + 1})^n + 1 = \lim_{n \to \infty} (\frac{n + 1 - 1}{n + 1})^n(\frac{n + 1 - 1}{n + 1})$$

8: $$\displaystyle \lim_{n \to \infty} (\frac{n }{n + 1})^n(\frac{n }{n + 1})$$

But the proof I've seen has substituted in the 'n' on it's own, not the 'n + 1' as I have & I don't understand, is there something I'm missing? Obviously the thing I've ended up with is strange, it looks similar to the original e^-1 but with that extra fraction.

#### Plato

MHF Helper
You may find this thread useful.
My reply there is a different approach.

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#### gerardhoyle

Well, that proof using L'Hopital's Rule is awesome, really smart!

Your inequality is good, I understand it but I have questions about it. Maybe I could ask you about it in a second.

As for this proof I've presesnted, I really need to understand what is going on in the last two parts! When the derivation for e to any arbitrary power of x comes in the author pulls the same kind of stunt!

Why does everything go astray when I put the 'n' back in?

I know that the minus 1 inside the brackets is what we're looking for but what is wrong about me re-inserting the 'n'?

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