- #1

Bashyboy

- 1,421

- 5

## Homework Statement

I am trying to show that ##n^{1/n}## is monotonically decreasing for ##n \ge 3##.

## Homework Equations

## The Attempt at a Solution

I am trying to prove the claim using induction. The base case is involves a trivial calculation. What I am having trouble is the induction step; i.e., assuming that ##n^{1/n} > (n+1)^{\frac{1}{n+1}}## is true, I want to show ##(n+1)^{\frac{1}{n+1}} > (n+2)^{\frac{1}{n+2}}##.

I have written the original inequality as ##n^{n+1} > (n+1)^n## and ##n > \left(\frac{n+1}{n} \right)^n## and working with these, but i have had no success. I could use a hint.