Is it true that $ e $ is equal to this?

Question

Is it true that

$$ e^{ab} = \lim_{n\to \infty} \Bigl(1 + \frac{a}{n}\Bigr) ^ {bn} $$

I remember seeing that forumla somewhere, and wasn't sure it was correct. Or if I don't remember it correctly.

If it is correct, can anyone send me a reference to a proof? or something that shows why it is correct

Answer 1

By definition it is $$e^{a} = \lim_{n\to \infty} \Bigl(1 + \frac{a}{n}\Bigr) ^ {n}.$$ And thus

$$\lim_{n\to \infty} \Bigl(1 + \frac{a}{n}\Bigr) ^ {bn}=\lim_{n\to \infty} \Bigl(\Bigl(1 + \frac{a}{n}\Bigr) ^ {n}\Bigr)^b=\Bigl(\lim_{n\to \infty} \Bigl(\Bigl(1 + \frac{a}{n}\Bigr) ^ {n}\Bigr)^b=(e^a)^b=e^{ab}.$$

Answer 2

Hint : Try to expand $e^{ab}$ by Taylor's Series Expansion. See what you've got !!