<s>[INST] Prove that $f(n) =\frac{(n+1)^n}{n^{n+1}}$ is Monotonic 
Prove that for each $n \in N$,
  $$f(n) =\frac{(n+1)^n}{n^{n+1}}$$
  is monotonic.

First, I can tell that the function is decreasing. If I take $\frac{1}{n}$, the function looks like $\frac{1}{n}(1+\frac{1}{n})^n$. Can this help?
 [/INST] Consider $$\frac{f(n)}{f(n+1)} = \frac{(n+1)^n}{n^{n+1}}\cdot \frac{(n+1)^{n+2}}{(n+2)^{n+1}}$$
$$ = \frac{(n+1)^n}{n^n \cdot n}\cdot \frac{(n+1)^n \cdot (n+1)^2}{(n+2)^n \cdot (n+1)}$$
$$ = \bigg(\frac{(n+1)^2}{n(n+2)}\bigg)^n \cdot\frac{(n+1)}{n}$$
Now for $n>0$ we have $$\frac{(n+1)^2}{n(n+2)}>1$$ And clearly $$\frac{n+1}{n} >1$$
So $$\frac{f(n)}{f(n+1)} >1 \Rightarrow f(n)>f(n+1)$$
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