If `L=lim_(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^(m)+4^(m)+….+n^(2m)))` is non zer finite real, then

Options

A. `L=3(e-1)`
B. `L=2(e-1)`
C. `m=1//3`
D. `m=1//3`

Answer

A, C

Explanation

`L=lim_(n to oo) (n^(3)sum_(r=1)^(n)e^(r//n))/((n+1)^(m)sum_(r=1)^(n)r^(2m))`
`=lim_(n to oo) (n^(3)sum_(r=1)^(n)e^(r//n) . 1/n)/((n+1)^(m)n^(2m)sum_(r=1)^(n)(r/n)^(2m) . 1/n)`
`=lim_(n to oo) (n^(3))/((n+1)^(m)n^(2m)) . (lim_(nto oo) 1/n sum_(r=1)^(n)e^(r//n))/(lim_(nto oo) 1/n sum_(r=1)^(n)(r/n)^(2m))`
`=lim_(nto oo) (n^(3))/((n^(3)+n^(2))m) . (int_(0)^(1)e^(x)dx)/(int_(0)^(1)x^(2m)dx)`
For `L` to be non-zero finite `m=1`
`:. L=(int_(0)^(1)e^(x)dx)/(int_(0)^(1)x^(2)dx)=(e-1)/(1//3) =3(e-1)`