<s>[INST] inequality with sum of powers How to prove the following inequality:
$$\forall n\geqslant 4:\dfrac {3^{n}+4^{n}+\cdots +\left( n+2\right) ^{n}} {\left( n+3\right) ^{n}} < 1$$
 [/INST] Induction base: verify for $n=4$:
$$
   \underbrace{3^4 + 4^4 + 5^4 + 6^4}_{2258} < 7^4 = 2401
$$
Induction step: Assuming inequality true for $n$, prove it true for $n+1$:
$$ \begin{eqnarray}
  \sum _{k=3}^{n+3} k^{n+1} &=& \sum _{k=3}^{n+2} k^{n+1}+(n+3)^{n+1} \leqslant (n+2)\sum _{k=3}^{n+2} k^{n}+(n+3)^{n+1}  \\
   &\stackrel{\text{by assumption}}{\leqslant}& (n+2)
   (n+3)^n+(n+3)^{n+1} \\ & < & 2 (n+3)^{n+1}<(n+4)^{n+1}
\end{eqnarray}
$$
The last inequality follows because the sequence $a_n = \left( 1 + \frac{1}{n+3} \right)^{n+1}$ is increasing (proof of Brian M. Scott, given in comments):
$$\begin{eqnarray}
    \frac{a_{n+1}}{a_n} &=& \left(1+\frac{1}{n+4}\right) \left( 1 - \frac{1}{(n+4)^2} \right)^{n+1} \\ &\stackrel{\text{binomial theorem}}{>}& \left(1+\frac{1}{n+4}\right) \left( 1 - \frac{n+1}{(n+4)^2} \right) = 1 + \frac{2n+11}{(n+4)^3} > 1
\end{eqnarray}$$ 
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