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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.

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  • $\begingroup$ Err... couldn't you answer this yourself? $\endgroup$
    – Pedro
    Aug 20, 2013 at 4:05
  • $\begingroup$ @PeterTamaroff I could have. $\endgroup$
    – Potato
    Aug 20, 2013 at 4:05
  • $\begingroup$ So, you're sharing? $\endgroup$
    – Pedro
    Aug 20, 2013 at 4:06
  • $\begingroup$ @PeterTamaroff Yes. $\endgroup$
    – Potato
    Aug 20, 2013 at 4:06
  • 1
    $\begingroup$ This is also given in Larson's Problem solving through problems as problem 7.1.15. $\endgroup$ Aug 20, 2013 at 7:26

1 Answer 1

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

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