How can one prove that $\lim_{n\to\infty}(1 +\frac k n)^n = e^k$?

Question

The equality in question is widely used to solve for limits like $\lim_{n\to\infty} \frac{n+1}{n-2}$, but how do you actually prove that
$$\lim_{n\to\infty} \left(1+\frac{k}{n}\right)^n = e^k.$$ Is it possible to prove it using the other formula: $\lim(1+\frac 1 n)^n = e?$

EDIT: I think I already got it: $$(1+ \frac{k}{n})^n = (1+ \frac{1}{\frac{n}{k}})^{k \cdot \frac{n}{k}} \rightarrow e^k$$ Is this correct?

Why would you use this limit for your example? Your limit for $(n+1)/(n-2)$ is trivial. — MrYouMath, Nov 21 '17 at 18:45
There are many questions that have answers to this very question: this answer shows that the series for $e^k$ equals this limit. — robjohn, Nov 21 '17 at 18:49
And also $\lim_{n\rightarrow \infty}(1+\frac{r}{n})^n$ is equal to ${e^{r}}$? — Sil, Nov 21 '17 at 18:49

score 3 · Answer 1 · answered Nov 21 '17 at 18:45

3

Since $f(x)=x^k$ is a continuous function, we obtain:

$$\lim_{n\to\infty}\left(1 +\frac k n\right)^n = \lim_{n\to\infty}\left(1 +\frac k n\right)^{\frac{n}{k}\cdot k } = \left(\lim_{n\to\infty}\left(1 +\frac k n\right)^{\frac{n}{k} }\right)^k=e^k$$

answered Nov 21 '17 at 18:45

Michael Rozenberg

194,933

How can one prove that $\lim_{n\to\infty}(1 +\frac k n)^n = e^k$?

1 Answers1