If f( x )= x +( x 2/1 !)+( x 3/2 !)+ ldots ldots ( x n /( n -1) !), then f(0)+f prime(0)+f prime prime(0)+ ldots ldots . .+f prime prime prime prime ldots . . . n text ntimes (0) is equal to -

Question

If f( x )= x +( x 2/1 !)+( x 3/2 !)+ ldots ldots ( x n /( n -1) !), then f(0)+f prime(0)+f prime prime(0)+ ldots ldots . .+f prime prime prime prime ldots . . . n text ntimes (0) is equal to - (A) ( n ( n +1)/2) (B) (( n 2+1)/2) (C) (( n ( n +1)/2))2 (D) (n(n+1)(2 n+1)/6)

Tardigrade Admin · Accepted Answer

Correct answer is (a) ( n ( n +1)/2)

Q. If $f (x) = x + \frac{x ^{2}}{1 !} + \frac{x ^{3}}{2 !} + \dots\dots \frac{x ^{n}}{( n - 1 )!}$ , then $f (0) + f^{'} (0) + f^{''} (0) + \dots\dots .. + f^{'''' \dots ... n ntimes} (0)$ is equal to -

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

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

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

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

Correct answer is (a) $\frac{n ( n + 1 )}{2}$

Q. If f(x)=x+1!x2​+2!x3​+……(n−1)!xn​, then f(0)+f′(0)+f′′(0)+……..+f′′′′…...n ntimes (0) is equal to -

2n(n+1)​

2(n2+1)​

(2n(n+1)​)2

6n(n+1)(2n+1)​