TITLE: Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$
QUESTION [10 upvotes]: How do I prove that:
$$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$
Do I use induction?

REPLY [5 votes]: You can use integral:
$$\frac { 1 }{ \sqrt { 1 }  } +\frac { 1 }{ \sqrt { 2 }  } +\dots +\frac { 1 }{ \sqrt { 100 }  } <\int _{ 0 }^{ 100 }{ \frac { 1 }{ \sqrt { x }  }  } dx=20$$
You can imagine approximating the integral with rectangles of side $\frac { 1 }{ \sqrt { n }  }$ and $1$, will give less area than the integral because of the behaviour of the curve.