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Popular Calculus >

taylor 1/x ,\at 1

Solution

taylor

\frac{1}{x}, at 1

Solution

\frac{- \frac{1}{( at 1 ) ^{2}}}{1 !} (x - at 1) + \frac{\frac{2}{( at 1 ) ^{3}}}{2 !} (x - at 1)^{2} + \frac{- \frac{6}{( at 1 ) ^{4}}}{3 !} (x - at 1)^{3} + \frac{\frac{24}{( at 1 ) ^{5}}}{4 !} (x - at 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{1}{at 1} + \frac{- \frac{1}{( at 1 ) ^{2}}}{1 !} (x - at 1) + \frac{\frac{2}{( at 1 ) ^{3}}}{2 !} (x - at 1)^{2} + \frac{- \frac{6}{( at 1 ) ^{4}}}{3 !} (x - at 1)^{3} + \frac{\frac{24}{( at 1 ) ^{5}}}{4 !} (x - at 1)^{4} + \dots

Refine

= \frac{- \frac{1}{( at 1 ) ^{2}}}{1 !} (x - at 1) + \frac{\frac{2}{( at 1 ) ^{3}}}{2 !} (x - at 1)^{2} + \frac{- \frac{6}{( at 1 ) ^{4}}}{3 !} (x - at 1)^{3} + \frac{\frac{24}{( at 1 ) ^{5}}}{4 !} (x - at 1)^{4} + \dots

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