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Question: If f(x) = x + \(\frac{x^{2}}{1!} + \frac{x^{3}}{2!}\)+ …….\(\frac{x^{n}}{(n - 1)!}\), then f(0) + f...

If f(x) = x + x21!+x32!\frac{x^{2}}{1!} + \frac{x^{3}}{2!}+ …….xn(n−1)!\frac{x^{n}}{(n - 1)!}, then

f(0) + f ′(0) + f ′′(0) + ….. f ′′′…….n times(0) is equal to

A

n(n+1)2\frac{n(n + 1)}{2}

B

n2+12\frac{n^{2} + 1}{2}

C

(n(n+1)2)2\left( \frac{n(n + 1)}{2} \right)^{2}

D

n(n+1)(2n+1)6\frac{n(n + 1)(2n + 1)}{6}

Answer

n(n+1)2\frac{n(n + 1)}{2}

Explanation

Solution

f ′(x) = 1 + 2x + 3x22!\frac{3x^{2}}{2!}…….

⇒ f ′(0) = 1

f " (x) = 2 + 3x + ……..

⇒ f "(0) = 2

f ′′′…….n times (0) = n

f(0) + f ′(0) + f "(0) + ……..+ f "" …..n times(0)
= 1 + 2 + 3 + …..+ n = n(n+1)2\frac{n(n + 1)}{2}