# An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$

Prove that for any positive integer $n$,

$$n^{1/n} < 1 + \sqrt{\frac{2}{n}}.$$

This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29.

Hint:

Use the binomial theorem.

• Err... couldn't you answer this yourself?
– Pedro
Aug 20, 2013 at 4:05
• @PeterTamaroff I could have. Aug 20, 2013 at 4:05
• So, you're sharing?
– Pedro
Aug 20, 2013 at 4:06
• @PeterTamaroff Yes. Aug 20, 2013 at 4:06
• This is also given in Larson's Problem solving through problems as problem 7.1.15. Aug 20, 2013 at 7:26

$$\left(1+\sqrt{\frac{2}{n}}\right)^n>1+C_n^2*\frac{2}{n}=n$$