taylor-1/x

Solution

taylor

- \frac{1}{x}

- 1 + (x - 1) - (x - 1)^{2} + (x - 1)^{3} - (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

= - 1 + \frac{\frac{d}{d x} ( - \frac{1}{x} ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( - \frac{1}{x} ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( - \frac{1}{x} ) ( 1 )}{3 !} (x - 1)^{3} + \dots

Evaluate Derivatives

= - 1 + \frac{1}{1 !} (x - 1) + \frac{- 2}{2 !} (x - 1)^{2} + \frac{6}{3 !} (x - 1)^{3} + \frac{- 24}{4 !} (x - 1)^{4} + \dots

Refine

= - 1 + (x - 1) - (x - 1)^{2} + (x - 1)^{3} - (x - 1)^{4} + \dots

What is taylor-1/x ?
The answer to taylor-1/x is -1+(x-1)-(x-1)^2+(x-1)^3-(x-1)^4+\ldots