If $s=1+\frac 12+\frac 13+......+\frac 1n$,then show that $(n+s)^s \gt n^n(n+1)$.

Question

I am stuck with the following problem:

If $s=1+\frac 12+\frac 13+......+\frac 1n$,then show that $(n+s)^s \gt n^n(n+1)$.

My try: Applying A.M. > G.M. [where A.M. means arithmatic mean and G.M. implies geometric mean] on the set $\{(1+1),(1+\frac 12),...................,(1+\frac 1n)\}$ we get , A.M.=$\frac {n+s}{n}$ and G.M. =$\{(1+1).(1+\frac 12)...................(1+\frac 1n)\}^{\frac 1n}$.

Then I am stuck. Can someone help?

Answer 1

You're pretty much there and applied the right idea.

The GM telescopes. It's ${\left(\frac{2}{1}.\frac{3}{2}.\frac{4}{3}...\frac{n+1}{n}\right)}^{\frac{1}{n}}=(n+1)^{\frac{1}{n}}$.

Presumably $n>1$.

So therefore (noting equality cannot hold as the terms are not equal)

$$\frac{n+s}{n}\gt (n+1)^{\frac{1}{n}}$$ $\implies$ $$n+s\gt n(n+1)^{\frac{1}{n}}$$ The LHS and RHS are positive, so we can exponentiate both sides by $n$. $$(n+s)^n\gt n^n(n+1)$$