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

If you already know about the exponential function, evaluate this as an indeterminate form $1^\infty$. Your calculus textbook should discuss such indeterminate forms. — GEdgar, Aug 14 '18 at 16:25
The limit definition of $e$ is mentioned in its Wikipedia page. It's discussed in this question, and also proved here. — Bilbottom, Aug 14 '18 at 16:26

score 2 · Answer 1 · answered Aug 14 '18 at 16:25

2

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

answered Aug 14 '18 at 16:25

Anik Bhowmick

1,051

score 1 · Accepted Answer · answered Aug 14 '18 at 16:26

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

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

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

2 Answers2