Define a sequence \{{s}_{n}\} of real numbers by {s}_{{n}}=\\sum_{{k}=0} ^{{n}} {1}/{\\sqrt{{n}^{2}+{k} }} \text{ for } {n} \\geq1Then {\lim } _{n \\to\infty} {s}_{{n}}

Correct Answer is : exists and lies in the interval [1, 2)

exists and lies in the interval (0,1)

exists and lies in the interval [1, 2)

exists and lies in the interval [2, ∞)

KVPY 2020-21 SB SX Exam Previous Papers

Question : 13 of 120

Marks: +1, -0

Define a sequence

\{{s}_{n}\}

of real numbers by

{s}_{{n}}=\∑↙{{k}=0} ↖{{n}} {1}/{\√{{n}^{2}+{k} }} \text" for " {n} \≥1

Then

{\lim } ↙{n \→\∞} {s}_{{n}}

Solution:

Given,

S_{n}=∑↙{k=0}↖{n} \;{1}/{√{n^{2}+k}}

⇒ \;{1}/{√{n^{2}+n}}+\;{1}/{√{n^{2}+n}}+. . . .+\;{1}/{√{n^{2}+n}}

≤ \;{1}/{√{n^{2}}}+\;{1}/{√{n^{2}+1}}+. . .+\;{1}/{√{n^{2}+n}}

≤ \;{1}/{√{n^{2}}}+\;{1}/{√{n^{2}}}+. .+\;{1}/{√{n^{2}}}

⇒ \lim ↙{n → ∞} \;{n+1}/{√{n^{2}+n}} ≤ \lim ↙{n → ∞} S_{n} ≤ \lim ↙{n → ∞} \;{n+1}/{√{n^{2}}}

⇒ 1 ≤ \lim ↙{n → ∞} S_{n} ≤ 1 ⇒ \lim ↙{n → ∞} S_{n}=1

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