<s>[INST] Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$ How do I prove that:
$$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$
Do I use induction?
 [/INST] Prove the following claim using induction on $n$:
$$\sum_{k=1}^n \dfrac1{\sqrt{k}} < 2 \sqrt{n}$$
In the induction, you will essentially need to show that
$$2\sqrt{n} +\dfrac1{\sqrt{n+1}} < 2 \sqrt{n+1} \tag{$\star$}$$
To prove $(\star)$, note that
$$\sqrt{n} < \sqrt{n+1} \implies \sqrt{n} + \sqrt{n+1} <2 \sqrt{n+1} \implies \dfrac1{\sqrt{n+1}} < \dfrac2{\sqrt{n} + \sqrt{n+1}}$$
Multiplying and divding the right hand side by $(\sqrt{n+1} - \sqrt{n})$, we get
$$\dfrac1{\sqrt{n+1}} < \dfrac2{\sqrt{n} + \sqrt{n+1}}\cdot \dfrac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = 2({\sqrt{n+1} - \sqrt{n}})$$
which gives us  $(\star)$.
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