Prove that $2 < (1 + \frac{1}{n})^n$ for $n \geq 2$

Question

Actually, I need to prove this statement for $n \in \mathbb{N}$, thus I can use induction, but still I didn't figure out how to prove it.

Also, I tried to prove that $(1 + \frac{1}{n})^n$ is strictly increasing. I denoted $x_n = (1 + \frac{1}{n})^n$, and tried to evaluate $\frac{x_{n+1}}{x_n}$ but still it didn't give useful result.

So how to prove this statement?

Bernoulli inequality? And please ask one question per post, so proving that $(1+\frac 1n)^n$ is strictly increasing is something for another post, though I'm sure it's already been asked. — Mastrem, Sep 17 '21 at 09:36

score 3 · Answer 1 · answered Sep 17 '21 at 09:55

3

You can use binomial theorem: $$\left(1+\dfrac{1}{n}\right)^n=1+\dfrac{1}{n}\cdot n+\dfrac{1}{n^2}\cdot\dfrac{n(n-1)}{2}+\cdots>1+1=2$$

answered Sep 17 '21 at 09:55

MafPrivate

4,033

That's straightforward but clever idea. Thanks – RussianStranger Sep 17 '21 at 11:02

Alessio K · Accepted Answer · 2021-09-17T10:53:54.427

1

You can also apply Bernoulli's inequality, that is $(1+x)^{n}\geq 1+ nx$ for every integer $n\geq 0$ and real number $x\geq -1$. The inequality is strict if $n\geq 2 $ and $x\neq 0$.

edited Sep 17 '21 at 10:53

answered Sep 17 '21 at 10:44

Alessio K

10,599

That was useful for my increasingness proof. Thanks – RussianStranger Sep 17 '21 at 11:02
You're very welcome! :-) – Alessio K Sep 17 '21 at 11:02

Prove that $2 < (1 + \frac{1}{n})^n$ for $n \geq 2$

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