Lim 1^2+2^2+3^2+.........n^2 /n^3 n€Nn-->infinitySolve fast....plsDo

1.	Lim 1^2+2^2+3^2+.........n^2 /n^3 n€Nn-->infinitySolve fast....plsDon't spma.....I will mark as brainliest....
Answer» ANSWER: ${\large{\overbrace{\underbrace{\purple{your \: \: answer: \frac{1}{3} }}}}} \\ \\ ♡ dear ----$ Step-by-step EXPLANATION: $\bf\red{ {1}^{2} + {2}^{2} + {3}^{2} + ........ {n}^{2} } \\ \\ \bf\red{ \underline{principle \: of \: mathematical \: induction \implies \frac{n(n + 1)(2n + 1)}{6} }} \\ \\ \bf\ lim \: \frac{ {1}^{2} + {2}^{2} + {3}^{2} + ............ + {n}^{2} }{ {n}^{3} } \: \: \: \: \: (n \: belongs \: to \: N) \\ n\mapsto \infty \\ \\ \bf\ \implies \: lim \frac{n(n + 1)(2n + 1)}{6 {n}^{3} } \\ n \mapsto \infty \\ \\ \bf\ \implies \: \frac{1}{6} \: lim( \frac{n + 1}{n} )( \frac{2n + 1}{n} ) \\ n \mapsto \infty \\ \\ \bf\ \implies \frac{1}{6} \: lim[(1 + \frac{1}{n} )(2 + \frac{1}{n})] \\ n \mapsto \infty \\ \\ \bf\ \implies \frac{1}{6} (1 + 0)(2 + 0) \: \: \: \: \: \: \: \: \bf\underline{ [\therefore \: n \mapsto \infty \: \: \: \frac{1}{n} \mapsto0] }\\ \\ \bf\boxed{ \bigstar \implies \frac{1}{3} \: \: \bigstar } \\ \\ \\$ $\large\boxed{ \fcolorbox{red}{pink}{hope \: it \: help \: you}}$

Lim 1^2+2^2+3^2+.........n^2 /n^3 n€Nn-->infinitySolve fast....plsDon't spma.....I will mark as brainliest....

Answer»

${\large{\overbrace{\underbrace{\purple{your \: \: answer: \frac{1}{3} }}}}} \\ \\ ♡ dear *----*$

Step-by-step EXPLANATION:

$\bf\red{ {1}^{2} + {2}^{2} + {3}^{2} + ........ {n}^{2} } \\ \\ \bf\red{ \underline{principle \: of \: mathematical \: induction \implies \frac{n(n + 1)(2n + 1)}{6} }} \\ \\ \bf\ lim \: \frac{ {1}^{2} + {2}^{2} + {3}^{2} + ............ + {n}^{2} }{ {n}^{3} } \: \: \: \: \: (n \: belongs \: to \: N) \\ n\mapsto \infty \\ \\ \bf\ \implies \: lim \frac{n(n + 1)(2n + 1)}{6 {n}^{3} } \\ n \mapsto \infty \\ \\ \bf\ \implies \: \frac{1}{6} \: lim( \frac{n + 1}{n} )( \frac{2n + 1}{n} ) \\ n \mapsto \infty \\ \\ \bf\ \implies \frac{1}{6} \: lim[(1 + \frac{1}{n} )(2 + \frac{1}{n})] \\ n \mapsto \infty \\ \\ \bf\ \implies \frac{1}{6} (1 + 0)(2 + 0) \: \: \: \: \: \: \: \: \bf\underline{ [\therefore \: n \mapsto \infty \: \: \: \frac{1}{n} \mapsto0] }\\ \\ \bf\boxed{ \bigstar \implies \frac{1}{3} \: \: \bigstar } \\ \\ \\$

$\large\boxed{ \fcolorbox{red}{pink}{hope \: it \: help \: you}}$

Lim 1^2+2^2+3^2+.........n^2 /n^3 n€Nn-->infinitySolve fast....plsDon't spma.....I will mark as brainliest....

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