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\lim_{ n \to \infty }\frac{1}{n}\sum_{r=1}^{2n}\frac{r}{\sqrt{n^{2}+r^{2}}}

  • Option 1)

    1

  • Option 2)

    0

  • Option 3)

    \sqrt{5}-1

  • Option 4)

    2

 

Answers (1)

As we learnt 

 

Properties of Definite Integration -

 

\int_{a}^{a+x\, T}f\left ( x \right )dx= x\int_{0}^{T}f\left ( x \right )dx

 

(Formula)

- wherein

If  f\left ( x \right ) is periodic with period T and x\in N

 

 \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{r=1}^{2n}\frac{\frac{r}{n}}{\sqrt{1+\left ( \frac{r}{n} \right )^{2}}}= \int_{0}^{2}\frac{x}{\sqrt{1+x^{2}}}dx

=\left [ \sqrt{1+x^{2}} \right ]^{2}_{0}= \sqrt{5}-1

 


Option 1)

1

Option 2)

0

Option 3)

\sqrt{5}-1

Option 4)

2

Posted by

Vakul

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