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If f(x) =x + x^(2)/(2!) + x^(3)/(3!) + …...

If f(x) `=x + x^(2)/(2!) + x^(3)/(3!) + ……+x^(n)/((n-1)!)`, then `f(0) + f^(1)(0) + f^(2)(0) +…………+f^(n)(0)` is equal to:

A

`(n(n+1))/2`

B

`(n^(2)+1)/2`

C

`((n(n+1))/2)^(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
We have, `f(x) = x+ x^(2)/(1!) + x^(3)/(2!) + ……..+x^(n)/((n-1)!)`
`f(x) = 1+ (2x)/(1!) + (3x^(2))/(2!) +………..+x^(n)/((n-1)!)`
`f^(2)(x) = 2+ (3.2x)/(2!) +………+(n(n-1)x^(n-2))/((n-1)!)`
`f^(n)(x) = (n!)/((n-1)!)`
`rArr f(0) + f^(1)(0) + f^(2)(0) + ...........+fn(0)`
`=0 + 1+2 + 3 +........n = (n(n+1))/2`
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