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Popular >

taylor ln(x/(x+a)),1

Solution

taylor

ln (\frac{x}{x + a}), 1

Solution

- ln (1 + a) + \frac{a}{1 + a} (x - 1) - \frac{a ( 2 + a )}{2 ( 1 + a ) ^{2}} (x - 1)^{2} + \frac{a ( a ^{2} + 3 a + 3 )}{3 ( a + 1 ) ^{3}} (x - 1)^{3} - \frac{a ( a + 2 ) ( a ^{2} + 2 a + 2 )}{4 ( a + 1 ) ^{4}} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= - ln (1 + a) + \frac{\frac{a}{1 + a}}{1 !} (x - 1) + \frac{- \frac{a ( 2 + a )}{( 1 + a ) ^{2}}}{2 !} (x - 1)^{2} + \frac{\frac{2 a ( a ^{2} + 3 a + 3 )}{( 1 + a ) ^{3}}}{3 !} (x - 1)^{3} + \frac{- \frac{6 a ( a + 2 ) ( a ^{2} + 2 a + 2 )}{( 1 + a ) ^{4}}}{4 !} (x - 1)^{4} + \dots

Refine

= - ln (1 + a) + \frac{a}{1 + a} (x - 1) - \frac{a ( 2 + a )}{2 ( 1 + a ) ^{2}} (x - 1)^{2} + \frac{a ( a ^{2} + 3 a + 3 )}{3 ( a + 1 ) ^{3}} (x - 1)^{3} - \frac{a ( a + 2 ) ( a ^{2} + 2 a + 2 )}{4 ( a + 1 ) ^{4}} (x - 1)^{4} + \dots

Popular Examples

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derivative of (sqrt(x)/(sin(x)))

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integral of (x^2)/(4x^2+9)

\int \frac{x ^{2}}{4 x ^{2} + 9} d x

Frequently Asked Questions (FAQ)

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