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Lim(n->oo)[1/n^2 * sec^2 (1/n^2)+2/n^2 *...

`Lim_(n->oo)[1/n^2 * sec^2 (1/n^2)+2/n^2 * sec^2 (4/n^2)+..............+1/n * sec^2 1]`

Text Solution

Verified by Experts

The correct Answer is:
`1/2 tan1`
`lim_(n to oo) [1/(n^(2)) "sec"^(2)1/(n^(2))+2/(n^(2))"sec"^(2) 4/(n^(2))+……..+1/nsec^(2)1]`
`=lim_(nto oo) 1/n sum_(r=1)^(n)r/n"sec"^(2)(r/n)^(2)`
`=int_(0)^(1)x sec^(2) x^(2) dx`
Put `x^(2)=t` so that `2x dx=dt`
When `x=0, t=0`, When `x=1, t=1`
`:. `Required limit `=1/2 int_(0)^(1) sec^(2) t dt `
`=1/2 [tan t]_(0)^(1)=1/2[tan 1-0]`
`=1/2 tan 1`
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