Help me please to understand when the inequality true.
Let $n
For which $n$ and $N$ the following is true $ n^{2n+1}\leq N^{N+1}? $
Thank you.
Help me please to understand when the inequality true.
Let $n
For which $n$ and $N$ the following is true $ n^{2n+1}\leq N^{N+1}? $
Thank you.
The inequality holds if $n\leq N^{1/2}$ and if $N^{1/2}\leq N-\frac12$:
Assuming these hypotheses, we get $n^2\leq N$ and so $n^{2n}\leq N^{N^{1/2}}\leq N^{N-\frac12}.$ Multiplying through once more by $n$, $n^{2n+1}\leq nN^{N-\frac12}\leq N^{1/2}N^{N-\frac12}=N^N.$
Since $N$ is a natural number, $N^{1/2}\leq N-\frac12$ if and only if $N\geq 4$. So we can say that the inequality is satisfied IF $n\leq\sqrt{N} \quad \hbox{and}\quad N\geq4.$ This is only a sufficient condition: I doubt that it is also necessary.
For $N=1,2,3$, we can check directly that only $n=1$ satisfies the inequality in those cases.