Integer part of a number

Let $\left \lfloor \cdot \right \rfloor$ denote the floor function. Evaluate the integer part of the number:

$$\left \lfloor \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ \cdots + \frac{1}{\sqrt{2010}} \right \rfloor$$

Solution

Using the double inequality

$$2\left ( \sqrt{n+1}- \sqrt{n} \right )< \frac{1}{\sqrt{n}} < 2 \left ( \sqrt{n} - \sqrt{n-1}\right )$$

and summing telescopically both the LHS and the RHS we get that the integer part of this number, call it $\mathcal{S}$ is $\left \lfloor \mathcal{S} \right \rfloor$=86