What is taylor 2/x ,3 ?

The answer to taylor 2/x ,3 is 2/3-2/9 (x-3)+2/27 (x-3)^2-2/81 (x-3)^3+2/243 (x-3)^4+\ldots

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taylor 2/x ,3

taylor

\frac{2}{x}, 3

\frac{2}{3} - \frac{2}{9} (x - 3) + \frac{2}{27} (x - 3)^{2} - \frac{2}{81} (x - 3)^{3} + \frac{2}{243} (x - 3)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{2}{3} + \frac{- \frac{2}{9}}{1 !} (x - 3) + \frac{\frac{4}{27}}{2 !} (x - 3)^{2} + \frac{- \frac{4}{27}}{3 !} (x - 3)^{3} + \frac{\frac{16}{81}}{4 !} (x - 3)^{4} + \dots

Refine

= \frac{2}{3} - \frac{2}{9} (x - 3) + \frac{2}{27} (x - 3)^{2} - \frac{2}{81} (x - 3)^{3} + \frac{2}{243} (x - 3)^{4} + \dots

What is taylor 2/x ,3 ?
The answer to taylor 2/x ,3 is 2/3-2/9 (x-3)+2/27 (x-3)^2-2/81 (x-3)^3+2/243 (x-3)^4+\ldots