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Popular >

taylor f(x)= x/(e^x-1)

Solution

taylor

f (x) = \frac{x}{e ^{x} - 1}

Solution

\frac{1}{e - 1} - \frac{1}{( e - 1 ) ^{2}} (x - 1) - \frac{e ^{2} - 3 e}{2 ( e - 1 ) ^{3}} (x - 1)^{2} + \frac{2 e ^{3} - 4 e ^{2} - 4 e}{6 ( e - 1 ) ^{4}} (x - 1)^{3} - \frac{3 e ^{4} + e ^{3} - 23 e ^{2} - 5 e}{24 ( e - 1 ) ^{5}} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{1}{e - 1} + \frac{- \frac{1}{( e - 1 ) ^{2}}}{1 !} (x - 1) + \frac{- \frac{e ^{2} - 3 e}{( e - 1 ) ^{3}}}{2 !} (x - 1)^{2} + \frac{\frac{2 e ^{3} - 4 e ^{2} - 4 e}{( e - 1 ) ^{4}}}{3 !} (x - 1)^{3} + \frac{- \frac{3 e ^{4} + e ^{3} - 23 e ^{2} - 5 e}{( e - 1 ) ^{5}}}{4 !} (x - 1)^{4} + \dots

Refine

= \frac{1}{e - 1} - \frac{1}{( e - 1 ) ^{2}} (x - 1) - \frac{e ^{2} - 3 e}{2 ( e - 1 ) ^{3}} (x - 1)^{2} + \frac{2 e ^{3} - 4 e ^{2} - 4 e}{6 ( e - 1 ) ^{4}} (x - 1)^{3} - \frac{3 e ^{4} + e ^{3} - 23 e ^{2} - 5 e}{24 ( e - 1 ) ^{5}} (x - 1)^{4} + \dots

Popular Examples

integral from 9 to 16 of 1/(8-sqrt(x))

\int_{9}^{16} \frac{1}{8 - x} d x

area y=x,y=(x-1)^2 area

y = x, y = (x - 1)^{2}

integral of ((ln(x))^{42})/x

\int \frac{( ln ( x ) ) ^{42}}{x} d x

integral of ((9+sqrt(x)+x))/x

\int \frac{( 9 + x + x )}{x} d x

derivative of (e^{2x}-5x^2)

\frac{d}{d x} ((e^{2 x} - 5 x)^{2})

Frequently Asked Questions (FAQ)

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