<s>[INST] Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$ How am I suppose to prove that:
$$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$$
Do I use the way like how we count $1+2+ \cdots+100$ to estimate?
So $1/5050 \lt 20$, implying that it is indeed less than $20$?
 [/INST] $\int^{100}_1 \frac{1}{\sqrt x} = 2\sqrt x|^{100}_1 = 2\times ( \sqrt 100 - \sqrt 1) = 18$
As, $\sum\limits^{100}_{1} \frac{1}{\sqrt x} < \int^{100}_1 \frac{1}{\sqrt x}$,
So, $\sum\limits^{100}_{1} \frac{1}{\sqrt x} < 20$
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