taylor 1/t

Solution

taylor

\frac{1}{t}

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

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

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

Refine

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

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