Question

Expand f(x) = ^1⁄x about x = 1.

(a) 1 – (x-1) + (x-1)^2 – (x-1)^3 + ….

(b) 1 + (x-1) + (x-1)^2 + (x-1)^3 + ….

(c) 1 + (x-1) – (x-1)^2 + (x-1)^3 + ….

(d) 1 – (x+1) + (x+1)^2 – (x+1)^3 + ….

The question was asked in examination.

I need to ask this question from Taylor Mclaurin Series in section Differential Calculus of Engineering Mathematics

Answer 1

The correct option is (a) 1 – (x-1) + (x-1)^2 – (x-1)^3 + ….

To explain: Given f(x) = ^1⁄x

Let, x – 1 = h

Hence, x = 1 + h

Hence, f(x) = f(1 + h) = f(1) + ^h⁄1! f’ (1) + ^h^2⁄2! f^” (1) +^h^3⁄3! f^”’ (1)+…

Now, f(1) = 1, f'(1) = -1, f”(1) = 2 ,f”'(1) = -6,…….

Hence, f(1 + h) = 1 – h + h^2 – h^3+….

hence, 1 – (x-1) + (x-1)^2 – (x-1)^3 +….