The value of `lim_(n to oo) ((1^(2)+2^(2)+………+n^(2)1^(3)+2^(3)+……….+n^(3)(1^(4)+2^(4)+…………n^(4)))/((1^(5)+2^(5)+…………+n^(5))^(2)))` is equal to

Options

A. `3/5`
B. `4/5`
C. `2/5`
D. `1/5`

Answer

A

Explanation

`lim_(n to oo) ((1/n)sum_(r=1)^(n)(r^(2))/(n^(2))(1/n sum_(r=1)^(n)(r^(3))/(n^(3)))(1/nsum_(r=1)^(n)(r^(4))/(n^(4))))/((1/n sum_(r=1)^(n)(r^(5))/(n^(5)))^(2))`
`=(int_(0)^(1)x^(2)dx. int_(0)^(1) x^(3) dx. int_(0)^(1) x^(4) dx)/((int_(0)^(1)x^(5)dx))`
`=(1/3 xx 1/4xx1/5)/(1/6xx1/6)=36/60=3/5`