Mathematics

# Evaluate: $\underset {n \rightarrow \infty} \lim \displaystyle \sum_{r=0}^{n-1}\frac{1}{n+r}$

$\log 2$

##### SOLUTION
$\lim\limits_{n \to \infty} \displaystyle\sum_{r=0}^{n-1} \dfrac{1}{n+r}$

$=\lim\limits_{n \to \infty} \dfrac{1}{n} \displaystyle\sum_{r=0}^{n-1} \dfrac{1}{\left(1+\dfrac{r}{n}\right)}$

Now, when $r=0, \dfrac{r}{n}= 0$
and when $r=n-1, \lim\limits_{n \to \infty} \dfrac{r}{n}=1$

Now, the given summation can be written as-
$\displaystyle\int_{0}^{1} \dfrac{1}{1+x} dx$

$=\left[\log{(1+x)}\right]_{0}^{1}$

$=\log{2}-\log{1}=\log2$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Single Correct Medium
Evaluate: $\displaystyle \int { x\tan ^{ -1 }{ x } dx }$
• A. $\displaystyle \frac { { x }^{ 2 } }{ 2 } \tan ^{ -1 }{ x } +\frac { 1 }{ 2 } \left[ x-\tan ^{ -1 }{ x } \right] +c$
• B. $\displaystyle \frac { { x }^{ 2 } }{ 2 } \tan ^{ -1 }{ x } +\frac { 1 }{ 2 } \left[ x+\tan ^{ -1 }{ x } \right] +c$
• C. none of these
• D. $\displaystyle \frac { { x }^{ 2 } }{ 2 } \tan ^{ -1 }{ x } -\frac { 1 }{ 2 } \left[ x-\tan ^{ -1 }{ x } \right] +c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of the definite integral $\int_\limits{ 1}^e( (x+1)e^x\ln x) dx$ is-
• A. $e$
• B. $e^e(e-1)$
• C. $e^{x+1}$
• D. $e^{e+1}+e$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle\int^{\pi/4}_{\pi/3}(\tan x+\cot x)^2dx$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $\displaystyle \frac{\mathrm{x}^{4}+24\mathrm{x}^{2}+28}{(\mathrm{x}^{2}+1)^{3}}$
$=\displaystyle \frac{\mathrm{A}\mathrm{x}+\mathrm{B}}{\mathrm{x}^{2}+1}+\frac{\mathrm{C}\mathrm{x}+\mathrm{D}}{(\mathrm{x}^{2}+1)^{2}}+\frac{\mathrm{E}\mathrm{x}+\mathrm{F}}{(\mathrm{x}^{2}+1)^{3}}$ then $A=$
• A. $1$
• B. $-1$
• C. $2$
• D. $0$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$