<s>[INST] Prove $(\frac{n+1}{n})^{n+1}$ is decreasing I managed to prove a similar fact: the following sequence is increasing: $\left( \dfrac{n+1}{n}\right)^n$, which means $\left( \dfrac{n+1}{n}\right)^n < \left( \dfrac{(n+1)+1}{(n+1)}\right)^{n+1}$.
All this took was some simple algebra and bernoulli's inequality. For the one in the title, I am not sure how.
My attempt was:
$$\left(\frac{n+1}{n}\right)^n \left(\frac{n+1}{n}\right)\stackrel{?}{>}\left(\frac{(n+1)+1}{(n+1)}\right)^{n+1} \left(\frac{(n+1)+1}{(n+1)}\right)$$
But this doesn't help me. Do you have other ideas?
 [/INST] Easy approach with A.M$\gt $G.M
Let $u_n=(\frac{n+1}{n})^{n+1}=\left(1+\frac{1}{n}\right)^{n+1}$
Let us consider n+2 positive numbers $(1-\frac{1}{n+1}),(1-\frac{1}{n+1}),(1-\frac{1}{n+1}),••••,(1-\frac{1}{n+1}),[(n+1)times] $and $1$.
Applying A.M$\gt $G.M.,we have,
$$\frac{(n+1)(1-\frac{1}{n+1})+1}{n+2}\gt (1-\frac{1}{n+1})^{\frac{n+1}{n+2}}$$
Or,$(\frac{n+1}{n+2})^{n+2}\gt (\frac{n}{n+1})^{n+1}$
Or,$(\frac{n+1}{n})^{n+1}\gt (\frac{n+2}{n+1})^{n+2}$
That is $ u_n\gt u_{n+1}$ for all $n$
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