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Popular Calculus >

taylor 1/(e^{1/x+1)-1}

Solution

taylor

\frac{1}{e ^{\frac{1}{x} + 1} - 1}

Solution

\frac{1}{e ^{2} - 1} + \frac{e ^{2}}{( e ^{2} - 1 ) ^{2}} (x - 1) - \frac{e ^{4} - 3 e ^{2}}{2 ( e ^{2} - 1 ) ^{3}} (x - 1)^{2} - \frac{- e ^{6} + 8 e ^{4} - 13 e ^{2}}{6 ( e ^{2} - 1 ) ^{4}} (x - 1)^{3} - \frac{- e ^{14} - 8 e ^{12} + 91 e ^{10} - 288 e ^{8} + 413 e ^{6} - 280 e ^{4} + 73 e ^{2}}{24 ( e ^{2} - 1 ) ^{8}} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{1}{e ^{2} - 1} + \frac{\frac{e ^{2}}{( e ^{2} - 1 ) ^{2}}}{1 !} (x - 1) + \frac{- \frac{e ^{4} - 3 e ^{2}}{( e ^{2} - 1 ) ^{3}}}{2 !} (x - 1)^{2} + \frac{- \frac{- e ^{6} + 8 e ^{4} - 13 e ^{2}}{( e ^{2} - 1 ) ^{4}}}{3 !} (x - 1)^{3} + \frac{- \frac{- e ^{14} - 8 e ^{12} + 91 e ^{10} - 288 e ^{8} + 413 e ^{6} - 280 e ^{4} + 73 e ^{2}}{( e ^{2} - 1 ) ^{8}}}{4 !} (x - 1)^{4} + \dots

Refine

= \frac{1}{e ^{2} - 1} + \frac{e ^{2}}{( e ^{2} - 1 ) ^{2}} (x - 1) - \frac{e ^{4} - 3 e ^{2}}{2 ( e ^{2} - 1 ) ^{3}} (x - 1)^{2} - \frac{- e ^{6} + 8 e ^{4} - 13 e ^{2}}{6 ( e ^{2} - 1 ) ^{4}} (x - 1)^{3} - \frac{- e ^{14} - 8 e ^{12} + 91 e ^{10} - 288 e ^{8} + 413 e ^{6} - 280 e ^{4} + 73 e ^{2}}{24 ( e ^{2} - 1 ) ^{8}} (x - 1)^{4} + \dots

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