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Understand the problem

For $$n\geq 1$$, the sequence $$\{x_n\}$$, where:
$$x_n=1+ \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}}+\cdots+ \frac{1}{\sqrt{n}}-2-2\sqrt{n}$$
is (a)decreasing
(b)increasing
(c)constant
(d)oscillating

Source of the problem

TIFR GS 2019, Part A, Problem 2

Series

Moderate

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Do you really need a hint? Try it first!

$$x_n=1+ \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}}+\cdots+ \frac{1}{\sqrt{n}}-2-2\sqrt{n}$$

$$x_{n+1}-x_n=\frac{2(\sqrt{n^2+n}-n)-1}{\sqrt{n+1}}$$

$$2(\sqrt{n^2+n}-n)-1<0$$

$$(\sqrt{n}+\frac{1}{2\sqrt{n}})^2=n+1+\frac{1}{4n}>n+1$$

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