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1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

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# Homework Help: Induction, tough one

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