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

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Discussion Overview

The discussion revolves around finding the highest element of the set A={\sqrt[n]{n}:n\in{N}}. Participants explore methods to determine the maximum value of the expression n^(1/n), considering both theoretical and practical approaches.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant asks how to find the highest element of the set A={\sqrt[n]{n}:n\in{N}}.
  • Another suggests examining the function \sqrt[x]{x} for x≥1 to find its absolute maximum and then correlating it to the corresponding n.
  • A hint is provided to locate the maximum value for real number arguments, noting that the function is increasing before the maximum and decreasing afterward, which allows for comparison of integer values around the maximizing number.
  • A participant questions whether it is possible to find the maximum value without using derivatives.
  • Another proposes comparing (n+1)^n and n^(n+1), indicating that the inequality changes after a certain number.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the methods to find the maximum value, and multiple approaches and suggestions are presented without resolution.

Contextual Notes

Some methods suggested may depend on the use of calculus, while others focus on algebraic comparisons. The discussion does not resolve whether derivatives are necessary for finding the maximum.

rahl__
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how can I find the highest element of such set:
A={[tex]\sqrt[n]{n}:n\in{N}[/tex]} ?
 
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Suggestion: You could look at [itex]\sqrt[x]{x}, \ \ x\geq 1[/itex]

and find its absolute max, and then match the "corresponding" n.
 
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.
 
can i find where the maximum value is without using derivatives?
 
Try compering (n+1)^n and n^(n+1). The inequality canges after a certian number.
 

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