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The value of $\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+r}}$ equals:
A. $\dfrac{1}{3}$
B. $\dfrac{1}{2}$
C. $\dfrac{2}{3}$
D. 1

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Last updated date: 16th Jun 2024
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Answer
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Hint: The squeeze theorem states that if we define functions such that h(x) ≤ f(x) ≤ g(x) and if $\underset{x\to a}{\mathop{\lim }}\,h(x)=\underset{x\to a}{\mathop{\lim }}\,g(x)=L$ , then $\underset{x\to a}{\mathop{\lim }}\,f(x)=L$ .
The sum $\sum\limits_{r=1}^{n}{{{r}^{2}}}=\dfrac{n(n+1)(2n+1)}{6}$ .

Complete step-by-step answer:
Let’s say that ${{S}_{n}}=\sum\limits_{r=1}^{n}{\dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+r}}$ .
Since 0 < r ≤ n, we can say that:
 \[{{n}^{3}}+{{n}^{2}}+n\ge {{n}^{3}}+{{n}^{2}}+r\ge {{n}^{3}}\]
After reciprocal them, sign of inequality changes,
⇒ \[\dfrac{1}{{{n}^{3}}+{{n}^{2}}+n}\le \dfrac{1}{{{n}^{3}}+{{n}^{2}}+r}\le \dfrac{1}{{{n}^{3}}}\]
Multiply by r2, we get
⇒ \[\dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+n}\le \dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+r}\le \dfrac{{{r}^{2}}}{{{n}^{3}}}\]
Taking submission from r=1 to r=n,
⇒ \[\sum\limits_{r=1}^{n}{\dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+n}}\le \sum\limits_{r=1}^{n}{\dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+r}}\le \sum\limits_{r=1}^{n}{\dfrac{{{r}^{2}}}{{{n}^{3}}}}\]
On solving,
⇒ \[\dfrac{1}{{{n}^{3}}+{{n}^{2}}+n}\sum\limits_{r=1}^{n}{{{r}^{2}}}\le {{S}_{n}}\le \dfrac{1}{{{n}^{3}}}\sum\limits_{r=1}^{n}{{{r}^{2}}}\]
⇒ \[\dfrac{1}{{{n}^{3}}+{{n}^{2}}+n}\left[ \dfrac{n(n+1)(2n+1)}{6} \right]\le {{S}_{n}}\le \dfrac{1}{{{n}^{3}}}\left[ \dfrac{n(n+1)(2n+1)}{6} \right]\]
On applying the limits, we get:
⇒ \[\dfrac{1}{6}\underset{n\to \infty }{\mathop{\lim }}\,\left[ \dfrac{2{{n}^{3}}+3{{n}^{2}}+n}{{{n}^{3}}+{{n}^{2}}+n} \right]\le \underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}\le \dfrac{1}{6}\underset{n\to \infty }{\mathop{\lim }}\,\left[ \dfrac{2{{n}^{3}}+3{{n}^{2}}+n}{{{n}^{3}}} \right]\]
Dividing the numerator and the denominator by the highest power of n, we get:
⇒ \[\dfrac{1}{6}\underset{n\to \infty }{\mathop{\lim }}\,\left[ \dfrac{2+\tfrac{3}{n}+\tfrac{1}{{{n}^{2}}}}{1+\tfrac{1}{n}+\tfrac{1}{{{n}^{2}}}} \right]\le \underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}\le \dfrac{1}{6}\underset{n\to \infty }{\mathop{\lim }}\,\left[ \dfrac{2+\tfrac{3}{n}+\tfrac{1}{{{n}^{2}}}}{1} \right]\]
Now, as $n\to \infty ,\dfrac{1}{n}\to 0$ .
⇒ \[\dfrac{1}{6}\left[ \dfrac{2+0+0}{1+0+0} \right]\le \underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}\le \dfrac{1}{6}\left[ \dfrac{2+0+0}{1} \right]\]
⇒ \[\dfrac{1}{6}(2)\le \underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}\le \dfrac{1}{6}(2)\]
⇒ \[\dfrac{1}{3}\le \underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}\le \dfrac{1}{3}\]
Therefore, by using the squeeze theorem, \[\underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\dfrac{{{r}^{2}}}{{{n}^{3}}+{{n}^{2}}+r}}=\dfrac{1}{3}\] .
The correct answer option is A.

Note: The squeeze theorem is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed.