Question 152
a) Prove that $\frac{1}{\sqrt{n+1}} \lt 2(\sqrt{n+1} - \sqrt{n}) \lt \frac{1}{\sqrt{n}}$ for every $n \in \mathbf{N}^{*}$.
b) Prove that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \lt 2 \sqrt{n-1}$ for every $n \in \mathbf{N}, n \geq 2$.
b) Prove that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \lt 2 \sqrt{n-1}$ for every $n \in \mathbf{N}, n \geq 2$.