Proof by induction: $n^n\leq 3^nn!$ for any $n\geq1$

Asked Dec 09 '23 at 11:05

Active Dec 09 '23 at 11:29

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I am struggling a lot trying to prove by indiction this inequality:

$$n^n\leq 3^nn! \text{ for any } n\geq1.$$

Could you please help me? When I try to prove it for $n+1$, I find myself in this condition: $$(n+1)^{n+1}\leq 3(n+1)3^nn!.$$

But I’m not sure how to continue from there. Thanks in advance for any kind of help!

edited Dec 09 '23 at 11:29

Rócherz

3,976

asked Dec 09 '23 at 11:05

Gaudio

Please use MathJax. Here is a tutorial. First cancel $n!$ on both sides. Then you have $1\le 3^n$, which should be easy to show. – Dietrich Burde Dec 09 '23 at 11:10
You say $n!\leq 3^n n!$, but in your induction step you seem to be talking about $n^n\leq n!3^n$ instead. Which one is it? – Arthur Dec 09 '23 at 11:11
@Arthur truly sorry! I ment $n^n$ I edited it now – Gaudio Dec 09 '23 at 11:13
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I think, Arthur is right, and you need to show that $n^n\le 3^nn!$. Note that this is a multiple duplicate, see for example here, or better this duplicate. – Dietrich Burde Dec 09 '23 at 11:13
@DietrichBurde It doesn't seem like a duplicate to me. In the one you linked they explicitly asked to not solve it with induction. Instead I need to solve it with induction. Thanks for your reply! – Gaudio Dec 09 '23 at 11:17
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Then take this duplicate, which uses induction. Did you look for other induction proofs already on this site? – Dietrich Burde Dec 09 '23 at 11:19
@DietrichBurde i think that the induction step in that answer is wrong. He multiplies (n+1)*(n/2)^n instead of doing ((n+1)/2)^n+1 – Gaudio Dec 09 '23 at 11:33
@JoséCarlosSantos I think your link solves my issue but I'm struggling a bit understanding the demonstration. Isn't very straight forward but still thanks a lot! – Gaudio Dec 09 '23 at 11:44
@DietrichBurde I ment this answer. This demonstration looks wrong to me. – Gaudio Dec 09 '23 at 11:44
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Multiplying an equation with $(n+1)$ is not yet wrong (in the beginning). But you have the choice to take the best answer (there are some more). – Dietrich Burde Dec 09 '23 at 11:47

Proof by induction: $n^n\leq 3^nn!$ for any $n\geq1$

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