Mathematics

# If $f\left ( n \right )= \left [ \left ( n+1 \right )\left ( n+2 \right )...\left ( n+n \right ) \right ]^{1/n}$, then $\lim_{n\rightarrow \infty }f\left ( n \right )$ equals

$4/e$

##### SOLUTION
Let $\displaystyle \lim _{ n\rightarrow \infty }{ f\left( n \right) } =A=\lim _{ n\rightarrow \infty }{ { \left[ \left( 1+\frac { 1 }{ n } \right) \left( 1+\frac { 2 }{ n } \right) ...\left( 1+\frac { n }{ n } \right) \right] }^{ \dfrac { 1 }{ n } } }$

$\displaystyle \Rightarrow \log { A= } \lim _{ n\rightarrow \infty }{ \frac { 1 }{ n } } \sum _{ r=1 }^{ n }{ \log { \left( 1+\frac { r }{ n } \right) } } =\int _{ 0 }^{ 1 }{ \log { \left( 1+x \right) } } dx=\log { \frac { 4 }{ e } }$

$\displaystyle \Rightarrow A=\frac { 4 }{ e }$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Subjective Medium
For what $a < 0$ does the inequality $\displaystyle\, \int_{a}^{0}(3^{-2x} \, \, 2.3^{-x})dx \geqslant 0$ hold true?

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Solve :
$\displaystyle \int \dfrac{dx}{sinx + cosx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following:
$\displaystyle \int_{0}^{\dfrac{1}{2}} \dfrac{dx}{(1 + x^2) \sqrt{1 - x^2}}$ (Hint: let $x = sin \,\theta$)

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
What is $\int _{ 0 }^{ 1 }{ x{ \left( 1-x \right) }^{ 9 }dx }$ equal to?
• A. $\dfrac{1}{132}$
• B. $\dfrac{1}{148}$
• C. $\dfrac{1}{240}$
• D. $\dfrac{1}{110}$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$