How can one show that $\displaystyle{\lim_{n \to \infty}\left(2 + {1 \over n}\right)^n}$ ?.
I am trying to shoehorn the definition of $e$ somewhere but fruitlessly so.
Asked
Active
Viewed 179 times
4
-
11$$2^n \leqslant \left(2+\frac1n\right)^n \leqslant 3^n$$ – Daniel Fischer Jan 04 '14 at 19:54
-
$\large\verb\displaystyle$ is not allowed in the main title. – Felix Marin Jan 04 '14 at 20:10
-
@user119114 : how can one show that what? What are you trying to show? – Stefan Smith Jan 05 '14 at 04:52
4 Answers
19
$$\displaystyle \lim_{n \to \infty} \left(2 + \frac 1n\right)^n\ge\lim_{n \to \infty} 2^n=+\infty$$
Mikasa
- 67,374
6
$$\left(2+\frac{1}{n}\right)^n = 2^n \left(1 + \frac{1/2}{n}\right)^n \sim 2^n \sqrt{e}$$
TMM
- 9,976
1
Since $\frac{1}{n}$ will asymptotically approach 0, the limit inside the parenthesis will go to 2. In turn, the limit of $2^n$ as $n$ approaches infinity is infinity as that function increases without bound.
-
4Be very careful here with taking limits piecemeal. It doesn't cost you here, but it usually does. – Ted Shifrin Jan 04 '14 at 20:17
