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

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

is (a)decreasing

(b)increasing

(c)constant

(d)oscillating

TIFR GS 2019, Part A, Problem 2

Series

Moderate

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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