Solveeit Logo
Login

Question

Mathematics Question on Continuity and differentiability

If f(x)=xnf (x) = x^n, then the value of f(1)−f′(1)1!+f′′(1)2!−f"′(1)3!+⋯+(−1)nfn(1)n!f \left(1\right)-\frac{f '\left(1\right)}{1!}+\frac{f ''\left(1\right)}{2!}-\frac{f "'\left(1\right)}{3!}+\dots+\frac{\left(-1\right)^{n}f ^{n}\left(1\right)}{n!} is

A

2n2^n

B

2n−12^{n-1}

C

00

D

11

Answer

00

Explanation

Solution

LHS=1−n1!+n(n−1)2!−n(n−1)(n−2)3!+………LHS = 1-\frac{n}{1!}+\frac{n\left(n-1\right)}{2!}-\frac{n\left(n-1\right)\left(n-2\right)}{3!}+\dots\dots\dots =1−nC1+nC2−………=1-^{n}C_{1}+^{n}C_{2}-\dots\dots\dots =0.=0.