The value of ` lim_(x to pi//2){1^(sec^2x) +2^(sec^2x) +3^(sec^2x)+..........+n^(sec^(2)x)}^(cos^(2)x)`is

Options

A. 0
B. n
C. `oo`
D. `(n(n+1))/(2)`

Answer

B

Explanation

Let `y=sec^2x`. Then , `y to oo as xto (pi)/(2)`
`lim_(xto pi//2){1^(sec^(2)x)+2^(sec^(2)x)+3^(sec^(2)x)+....+n^(sec^(2)x)}^(cos^(2)x)`
`lim_(ytooo ){1^(y)+2^y+3y+......+n^y}^(1//y)`
` =lim_(xto oo) n{((1)/(n))^y+((2)/(n))^y+((3)/(n))^y+......+((n)/(n))^y}^(1//y)`
` n {0+0+......+1}^0=n`