Mathematics

# $\underset {n\rightarrow \infty}{lim}\dfrac{1^2+2^2+3^2+.....+n^2}{n^3}=.................$

##### SOLUTION
$\displaystyle \lim _{ n\rightarrow \infty }{ \dfrac { { 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }......{ n }^{ 2 } }{ { n }^{ 3 } } } \\ \displaystyle \lim _{ n\rightarrow \infty }{ \dfrac { \sum { { n }^{ 2 } } }{ { n }^{ 3 } } } \\\displaystyle \lim _{ n\rightarrow \infty }{ \dfrac { n(n+1)(2n+1) }{ { 6n }^{ 3 } } } \\ \lim _{ n\rightarrow \infty }{ \dfrac { { 2n }^{ 3 }+3{ n }^{ 2 }+n }{ { 6n }^{ 3 } } } \\ =\dfrac { 2 }{ 6 } =\dfrac { 1 }{ 3 }$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Multiple Correct Hard
If $\displaystyle \int \frac{x^{2}-x+1}{(x^{2}+1)^{\frac{3}{2}}}e^{x}dx=e^{x}f(x)+c,$ then

• A. f(x) has two points of extrema
• B. f(x) is an even function
• C. f(x) is a bounded function
• D. the range of f(x) is (0,1]

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral:
$\int { \cfrac { 1+\cos { x } }{ { \left( x+\sin { x } \right) }^{ 3 } } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate the following:
$\displaystyle \int\limits_{\pi /3}^{\pi /2} {{{(\tan x + \cot x)}^2}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve: $\displaystyle \overset{\frac{1}{2}}{\underset{\frac{1}{2}}{\int}} [x]dx$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.