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I am currently practicing series and I have come across this problem many times.

I know how to prove that $$\large n^{\frac{1}{n}}$$ is bounded by $1$ and by $2$.

What I don't know is how to prove its monotonicity for natural numbers without using derivatives and expansions.

Can anyone help?

l0ner9
  • 623
  • But it isn't monotone. –  Nov 22 '20 at 22:10
  • @Gae.S. How so? – l0ner9 Nov 23 '20 at 17:04
  • Because $\sqrt[4]4=\sqrt2< \sqrt[3]3$. –  Nov 23 '20 at 17:06
  • @Gae.S. That's a terribly incomplete and one-sided answer. More so because I never said over all natural numbers. – l0ner9 Nov 23 '20 at 17:19
  • "What I don't know is how to prove its monotonicity for natural numbers without using derivatives and expansions." –  Nov 23 '20 at 17:20
  • @Gae.S. Read it out loud one more time. If you get stuck I'll be around to re-word it into more simple terms. – l0ner9 Nov 23 '20 at 17:22
  • Monotonicity is inherently a global property: if you don't want it globally, you have to state it, rather than suggesting that others cannot read your precious pieces of wisdom. –  Nov 23 '20 at 17:22

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