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Popular >

taylor 1/(1-e^x)

Solution

taylor

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

Solution

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

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

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

Refine

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

Popular Examples

sum from n=0 to infinity of (3+2n)/(2^n)

n = 0 \sum \infty \frac{3 + 2 n}{2 ^{n}}

sum from n=0 to infinity of 1/(n+13)

n = 0 \sum \infty \frac{1}{n + 13}

integral of y/((y^2-4)^{2/5)}

\int \frac{y}{( y ^{2} - 4 ) ^{\frac{2}{5}}} d y

tangent of f(x)=12x-x^3 tangent of

f (x) = 12 x - x^{3}

integral of-cos(nx)

\int - cos (n x) d x

Frequently Asked Questions (FAQ)

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