Prove that $\left(\sum_{k=1}^n\,\left(\frac{k}{n}\right)^n\right)^{\frac{1}{n}}$ decreases as $n$ increases.

Please help me to prove the following problem with inductive argument $$\left(\left(\frac{1}{n+1}\right)^{n+1}+\dots+\left(\frac{n+1}{n+1}\right)^{n+1}\right)^{\frac{1}{n+1}}<\left(\left(\frac{1}{n}\right)^n+\dots+\left(\frac{n}{n}\right)^n \right)^{\frac{1}{n}}\,.$$

I think we can use the following solution to prove the main problem:

$$\frac{1}{n+1}<\frac1n$$
$$(\frac1{n+1})^{n+1}<(\frac1n )^n$$

Thanks...

• I tried to edit your post to make it more readable. Please check to make sure that I didn't mess anything up. – Michael Burr Jul 2 '18 at 10:04
• It looks like there might be better avenues to prove this than with induction. Is induction required? – Michael Burr Jul 2 '18 at 10:09
• Thanks Michael. Yes induction is not best answer but its required. – MH Hemmati M Jul 2 '18 at 10:11
• Oooppss, I made a computational error. In the comment above, $T_n$ should be replaced by $$T_n:=\frac{\text{e}}{\text{e}-1} \left(1-\frac{\text{e}+1}{2(\text{e}-1)^2n}\right)\,.$$ – Batominovski Jul 2 '18 at 15:49
• @Batominovski: $$S_n = \frac e{e-1}\left(1-\frac1{2n}\frac{e+1}{(e-1)^2}+\frac1{24n^2}\frac{-5e^3+9e^2+57e+11}{(e-1)^4}+O\!\left(\frac1{n^3}\right)\right)$$ Not that that makes things any easier. – robjohn Jul 2 '18 at 19:15