6hey no one tryin???:O
1. If f: [-1 , 1] -> R and f ' (0) = limit n tends to infinity nf (1/n) and f(0) = 0 . Find the value of lim n tends to infinity 2/pi (n+1) cos inverse (1/n) - n . Given that 0< mod ( limit n tends to infinity cos inverse (1/n) ) < pi/2.
2. Suppose p(x) = a0 + a1 (x) + a2 (x^2) +....................+ an x ^n . If mod of p(x) is less than equal to mod of( e ^x-1) - 1 for all x greater than equal to 0 , prove that mod of a1 + a2 + .........+ n(an) is less than equal to 1.
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10 Answers
1hey the last line of the question 2 is might be wrong
it must be a1+2 a2
1Lt 2/Ï€[(n+1)\cos^{-1} 1/n]-n
n->∞
put 1/n=x as n-->∞,x-->0
Lt 2/Ï€[(1+x)\cos^{-1} x] -- 1
x-->0 ---------------------------------------- ( 0/0 form)
x
applying L hosp
you get 1-2/Ï€