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For natural number $n\ge5$, by mathematical induction or otherwise, prove that $n^n<(n-1)^{n+1}$.

Actually I was trying to solve the problem that I posted.


My Attempt:

Step I: Verify that when $n=5$, then the given inequality holds true;

$5^5\overset{?}{<}4^6\Rightarrow 3125<4096$ which is true.

Step II: Assume that the inequality is true for $n=k\ge 5$;

that is $k^k<(k-1)^{k+1}$ is true for $k\ge5$.

Step III: We need to prove that if the inequality is true for $n=k$, then it is true for $n=k+1$ as well;

that is to show that if $k^k<(k-1)^{k+1}$ is true, then $(k+1)^{k+1}<k^{k+2}$ is also true.

I have a difficulty to complete this part of the proof. Any help would be appreciated.

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I'd write $m=n-1$. Then the statement reduces to $$\left(1+\frac1m\right)^m<\frac{m^2}{m+1}.$$ It's well-known that $(1+1/m)^m$ increases to $e$, but more naively, $$\left(1+\frac1m\right)^m=1+1+\frac1{m^2}{m\choose 2} +\frac1{m^3}{m\choose 3}+\cdots<1+1+\frac12+\frac16+\cdots<3.$$ But $$\frac{m^2}{m+1}>\frac{m^2-1}{m+1}=m-1>3$$ if $n>4$.

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  • $\begingroup$ $m-1>3\iff m>4\iff n>5$, which means that you need to treat the case $n=5$ separately. Or use that because $e<3$ already is a strict inequality, and you only need one strict inequality in a chain to get strict inequality from first to last term, it is sufficient to demand $m-1\ge 3\iff n\ge 5$. $\endgroup$ – Lutz Lehmann Jul 9 '19 at 6:55
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Here is a proof without induction.

$(1-1/n)^n$ is strictly increasing for $n > 1 $ and approaches $1/e$ as $n \to \infty$ whereas $1/(n-1)$ is strictly decreasing and approaches $0$ as $n \to \infty$. Hence there is an $N_0$ such that for all $n > N_0$,

$$ \Big(1 - \frac{1}{n}\Big)^n > \frac{1}{n-1} $$

By little computation, we find that smallest $n$ for which this inequality holds is $n = 5$. Since LHS is increasing and RHS is decreasing, it implies that the inequality will never be violated for $n \ge 5$.

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The claim $x^x<(x-1)^{x+1}$ is equivalent to $$ x\ln x<(x+1)\ln(x-1). $$ Consider $f(x)=(x+1)\ln(x-1)-x\ln x$ and prove that

  1. $f'(x)=\ln\left(1-\frac1x\right)+\frac2{x-1}\to 0$ as $x\to +\infty$,
  2. $f''(x)=-\frac{x+1}{x(x-1)^2}<0$ for $x>1$.

Then $f'(x)>0$ and $f$ increases. From $f(5)>0$ it follows that $f(x)>0$ for all $x\ge 5$.

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    $\begingroup$ Or you can rewrite it to $\frac{\ln x}{x+1} < \frac{\ln (x-1)}x$ and show that the function $g(x) = \frac{\ln x}{x+1}$ is decreasing. $\endgroup$ – Quang Hoang Jul 9 '19 at 18:43
  • $\begingroup$ @QuangHoang Yes, it is smart rewriting. $\endgroup$ – A.Γ. Jul 9 '19 at 19:04

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