Prove $\left(1 + 1/\sqrt{n}\right)^n > \sqrt{n}$ for all natural $n$

Question

$$\left(1 + 1/\sqrt{n}\right)^n > \sqrt{n}$$

I'm trying to use Bernoulli's inequality

So $\left(1 + 1/\sqrt{n}\right)^n \ge 1 + n/\sqrt{n}$, but I'm not sure what to do from there.

Could I say that $1 + n/\sqrt{n} > \sqrt{n}$ for all natural numbers so the above statement holds true?

$\frac{n}{\sqrt n} \times \frac{\sqrt n}{\sqrt n} = \sqrt n$ — Nicholas, Oct 30 '15 at 16:27

score 2 · Accepted Answer · answered Oct 30 '15 at 16:28

2

$$ 1 + \frac{n}{\sqrt{n}} = 1 + \frac{\sqrt{n} \cdot \sqrt{n}}{\sqrt{n}} = 1 + \sqrt{n} > \sqrt{n} $$

answered Oct 30 '15 at 16:28

gt6989b

1 Answers1