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
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