Mathematics

# $\displaystyle \lim_{n\rightarrow \infty }\sum_{r=1}^{n-1}\frac{1}{\sqrt{n^{2}-r^{2}}}$ is

$\displaystyle \pi /2$

##### SOLUTION
$\displaystyle \lim _{ n\rightarrow \infty } \sum _{ r=1 }^{ n-1 } \dfrac { 1 }{ \sqrt { n^{ 2 }-r^{ 2 } } } =\lim _{ n\rightarrow \infty } \sum _{ r=1 }^{ n-1 } \dfrac { 1 }{ n } \dfrac { 1 }{ \sqrt { 1-\dfrac { { r }^{ 2 } }{ { n }^{ 2 } } } }$

$\displaystyle =\lim _{ n\rightarrow \infty } \dfrac { 1 }{ n } \sum _{ r=1 }^{ n-1 } \dfrac { 1 }{ \sqrt { 1-\dfrac { { r }^{ 2 } }{ { n }^{ 2 } } } } =\int _{ 0 }^{ 1 }{ \dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } dx$

$\displaystyle ={ \left[ \sin ^{ -1 }{ x } \right] }_{ 0 }^{ 1 }=\dfrac { \pi }{ 2 } -0=\dfrac { \pi }{ 2 }$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium

Evaluate the following definite integral:

$\displaystyle\int_{0}^{2}x\sqrt{x+2}dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Integrate the following functions: $\int\sin(mx)dx$
• A. $\dfrac{\cot mx}{4m}+c$
• B. $\dfrac{\sin 3mx}{m}+c$
• C. $-\dfrac{\cos 4mx}{4m}+c$
• D. $-\dfrac{\cos mx}{m}+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle \int \frac{x^{3}dx}{\sqrt{1+x^{2}}}$ is equal to
• A. $\displaystyle \frac{1}{3}\sqrt{1+x^{2}}(2+x^{2})+C$
• B. $\displaystyle \frac{1}{3}\sqrt{1+x^{2}}(x^{2}-1)+C$
• C. $\displaystyle \frac{1}{3}(x^{2}-1)^{3/2}+C$
• D. $\displaystyle \frac{1}{3}\sqrt{1+x^{2}}(x^{2}-2)+C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Find an anti derivative (or integral) of the given function by the method of inspection. $e^{2x}$

$\int {\dfrac {\cos 2x}{\sin x}}dx$