$\ln n+\sqrt{\frac12}+\sqrt{\frac23}+\ldots +\sqrt{\frac{n-1}{n}}<\sqrt2+\sqrt{\frac32}+\sqrt{\frac43}+\ldots +\sqrt{\frac{n}{n-1}}$

Question

Prove for every $n>1$

$$\ln n+\sqrt{\frac12}+\sqrt{\frac23}+\ldots +\sqrt{\frac{n-1}{n}}<\sqrt2+\sqrt{\frac32}+\sqrt{\frac43}+\ldots +\sqrt{\frac{n}{n-1}}$$

Answer 1

Hint

To use an induction you need to prove that: $$\ln \left(\frac{n+1}{n} \right) +\sqrt{\frac{n}{n+1}} \leq \sqrt{\frac{n+1}{n}} $$

So it sufficient to prove that for $x>1$: $$\ln \left(x \right) +\sqrt{\frac{1}{x}} - \sqrt{x} <0$$ which is equivalent ($\xi=x^2$) to prove that for $\xi>1$: $$f(\xi)=2\ln \left(\xi \right) +\frac{1}{\xi} - \xi <0 $$

Notice that: $$f(1)=0$$ and for $\xi>1$: $$f'(\xi)=\frac{2}{\xi}-\frac{1}{\xi^2}-1=-\frac{\xi^2-2\xi-1}{\xi^2}=-\frac{(\xi-1)^2}{\xi^2}<0$$