New Forum Member: Induction Problem Solving

[kky]

just joined the forum

Homework Statement

n^(1/n) > (n+1)^1/(n+1) for all n>=3.

Homework Equations

The Attempt at a Solution

k^1/k > (k+1)^1/(k+1)

=> k > (k+1)^k/(k+1)
=> k+1 > (k+1)^k/(k+1) + 1
=> (k+1)^1/(k+1) > [(k+1)^k/(k+1) + 1]^1/(k+1)


I then tried binomial expansion of the term on the right.
leads to

(k+1)^1/(k+1) > (k+1)^k/(k+1)2 + 1/(k+1)*(k+1)^{[k/(k+1)][1/(k+1) -1]} + (-k)/(2(k+1)²)*(k+1)^{[k/(k+1)][1/(k+1) - 2]}...

But seem to be getting nowhere because of the negative term that appears and will continue to appear in every other term...
Am i on the right path?

apologies if it is too easy.

 

[kky]

and btw, I've figured out how to do it without using induction.
but i want to know how to prove it using induction