Comparing the value of $(1^r + 2^r +3^r +\ ...\ + n^r)^n$ with $(n^n)(n!)^r$, ${(n)^2n}{(n!)^r}$, ${(n)^n}{(n!)^2r}$, ${(n)^n}{(n!)^2r}$

Question

What will be my approach toward this type of question:

The value of $$(1^r + 2^r +3^r +\ ...\ + n^r)^n$$ where $n$ is a real number, is

A.greater then or equal to $(n^n)(n!)^r$

B.less than or equal to ${(n)^2n}{(n!)^r}$

C. less than ${(n)^n}{(n!)^2r}$

D.greater than $(n^n)(n!)^r$

What I would do is start to write some bounds for $(\sum_{k=1}^n k^r)^n$, by example $$\left(\sum_{k=1}^n k^r\right)^n\le n^{nr}$$ — 2017-02-25
Relevant questions: https://approach0.xyz/search/?q=%24(1%5Er%20%2B%202%5Er%20%2B%5C%20...%5C%20%2B%20n%5Er)%5En%24&p=1 — 2017-02-25

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