Highest element of the expression n^(1/n)

[rahl__]

how can I find the highest element of such set:
A={$$\sqrt[n]{n}:n\in{N}$$} ?

 

[quasar987]

Suggestion: You could look at $\sqrt[x]{x}, \ \ x\geq 1$

and find its absolute max, and then match the "corresponding" n.

 

[arildno]

Hint:
Find out where the maximum value is for real number arguments.
Since the function is increasing prior to the maximizing number, and decreasing afterwards, you only need to compare the function values of the integer just prior to the maximizing number and of the integer just after the maximizing number.

 

[rahl__]

can i find where the maximum value is without using derivatives?

 

[vaishakh]

Try compering (n+1)^n and n^(n+1). The inequality canges after a certian number.