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Popular >

taylor 1/(e^x-1)

Solution

taylor

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

Solution

\frac{1}{e - 1} - \frac{e}{( e - 1 ) ^{2}} (x - 1) - \frac{- e ^{2} - e}{2 ( e - 1 ) ^{3}} (x - 1)^{2} + \frac{- 4 e ^{2} - e ^{3} - e}{6 ( e - 1 ) ^{4}} (x - 1)^{3} - \frac{- 11 e ^{2} - e ^{4} - 11 e ^{3} - 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{1}{e ^{x} - 1} ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{e ^{x} - 1} ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{e ^{x} - 1} ) ( 1 )}{3 !} (x - 1)^{3} + \dots

Evaluate Derivatives

= \frac{1}{e - 1} + \frac{- \frac{e}{( e - 1 ) ^{2}}}{1 !} (x - 1) + \frac{- \frac{- e ^{2} - e}{( e - 1 ) ^{3}}}{2 !} (x - 1)^{2} + \frac{\frac{- 4 e ^{2} - e ^{3} - e}{( e - 1 ) ^{4}}}{3 !} (x - 1)^{3} + \frac{- \frac{- 11 e ^{2} - e ^{4} - 11 e ^{3} - e}{( e - 1 ) ^{5}}}{4 !} (x - 1)^{4} + \dots

Refine

= \frac{1}{e - 1} - \frac{e}{( e - 1 ) ^{2}} (x - 1) - \frac{- e ^{2} - e}{2 ( e - 1 ) ^{3}} (x - 1)^{2} + \frac{- 4 e ^{2} - e ^{3} - e}{6 ( e - 1 ) ^{4}} (x - 1)^{3} - \frac{- 11 e ^{2} - e ^{4} - 11 e ^{3} - e}{24 ( e - 1 ) ^{5}} (x - 1)^{4} + \dots

Popular Examples

tangent of 1/(2sqrt(x+1))tangent of

\frac{1}{2 x + 1}

integral of (2x)/((2x^2-3)^3)

\int \frac{2 x}{( 2 x ^{2} - 3 ) ^{3}} d x

integral of arctan(4t)

\int arctan (4 t) d t

volumeabout y=0,y=2x^2,y=0,[0,1]volume about

y = 0, y = 2 x^{2}, y = 0, [0, 1]

sum from n=3 to infinity of 1/(ln(n))

n = 3 \sum \infty \frac{1}{ln ( n )}

Frequently Asked Questions (FAQ)

What is taylor 1/(e^x-1) ?
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