If [x] denotes the greatest integer less than or equal to x, then evaluate `lim_(ntooo) (1)/(n^(3))([1^(2)x]+[2^(2)x]+[3^(2)x]+...+[n^(2)x]).`
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We have, `underset(ntooo)lim(sum_(r=1)^(n)[r^(2)x])/(n^(3))=underset(ntooo)lim(underset(r=1)overset(n)sumr^(2)x-{r^(2)x})/(n^(3))` Where `{.}` denotes the fractional part function `=underset(ntooo)lim((x.(n(n+1)(2n+1))/(6))/n^(3)-underset(r=1)overset(n)sum{{r^(2)x}}/(n^(3)))` `=x/6underset(ntooo)lim(1+(1)/(n))(2+(1)/(n))-underset(ntooo)limunderset(r=1)overset(n)sum{{r^(2)x}}/(n^(3))` `=x/6-0" "( :. 0le{r^(2)x}lt1)` `=x/6`
LINEAR COMBINATION OF VECTORS, DEPENDENT AND INDEPENDENT VECTORS
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