<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] A close value can be obtained easily.
Let's look at the squares until $100$:
$1, 4, 9, 16, 25, 36, 49, 64, 81$.
We then know that the sum will be smallar than:
$$3\cdot 1+ 5\cdot\frac{1}{2}+7\cdot\frac{1}{3}+9\cdot\frac{1}{4}+...=
\sum_{n=1}^9\frac{2n+1}{n}=\sum_{n=1}^9 2 +\frac{1}{n} =20.82$$
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