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Differential Calculus> If f(x)=lt n->infinite ([x^2]+[(2x)^2]+[(...

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If f(x)=lt_n->infinite([x^2]+[(2x)^2]+[(4x)^2]+...+[(nx)^2]/n^3,prove that f(x) is a continuous function.[] is GIF

Shadow , 10 Years ago

Grade 11

1 Answers

Arun Kumar

Last Activity: 10 Years ago

$f(x)=\lim_{n \to \infty}([x^2]+[(2x)^2]+[(4x)^2]+...+[(nx)^2])/n^3 \\=>f(x)=\lim_{n \to \infty}(x^2+(2x)^2+(4x)^2+...+(nx)^2]-(\left \{ x^2\right \}+\left \{ (2x)^2 \right \}..)))/n^3 \\=>f(x)=\lim_{n \to \infty}((2n+1)(n+1)n(x^2)/6-(\left \{ x^2\right \}+\left \{ (2x)^2 \right \}..))/n^3 \\=>f(x)=x^2/3$

The fractional part is

$\leqslant n/2*1$

because max value of {} is 1 so lim =0

Now its clear that the function is continuos.

Arun Kumar

IIT Delhi

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