- #26

- 217

- 1

[itex]\sum_{n=1}^{m+1} \frac 1{\sqrt{n}} = ( \sum_{n=1}^{m} \frac 1{\sqrt{n}}) + \frac 1{\sqrt{m+1}} [/itex]

Your problem is to show that

[tex] 2 \sqrt{n+1}-2 + \frac{1}{\sqrt{n+1}} \geq 2 \sqrt{n+2} - 2,\\

\text{or}\\

\sqrt{n+2} - \sqrt{n+1} \leq \frac{1}{2 \sqrt{n+1}}[/tex]

This inequality implies that 9>8. And that is certainly true, therefore the inequality that you posted must be true.