en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

taylor x/(e^x-1),1

Solution

taylor

\frac{x}{e ^{x} - 1}, 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 of (5-3x^2)ln(x)

\int (5 - 3 x^{2}) ln (x) d x

f(x)=arctan(x^2)

f (x) = arctan (x^{2})

derivative of (x^5)/5+1/(12x^3)derivative of

\frac{x ^{5}}{5} + \frac{1}{12 x ^{3}}

integral of (3x^5)/(sqrt(1-6x^6))

\int \frac{3 x ^{5}}{1 - 6 x ^{6}} d x

integral of 1/(sqrt(3x-x^2-2))

\int \frac{1}{3 x - x ^{2} - 2} d x

Frequently Asked Questions (FAQ)

What is taylor x/(e^x-1),1 ?
The answer to taylor x/(e^x-1),1 is