What is taylor ln(x+2)-1 ?

The answer to taylor ln(x+2)-1 is ln(2)-1+1/2 x-1/8 x^2+1/24 x^3-1/64 x^4+\ldots

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taylor ln(x+2)-1

taylor

ln (x + 2) - 1

ln (2) - 1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{24} x^{3} - \frac{1}{64} x^{4} + \dots

Solution steps

Apply the Taylor Formula

= ln (2) - 1 + \frac{\frac{d}{d x} ( ln ( x + 2 ) - 1 ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ln ( x + 2 ) - 1 ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ln ( x + 2 ) - 1 ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= ln (2) - 1 + \frac{\frac{1}{2}}{1 !} x + \frac{- \frac{1}{4}}{2 !} x^{2} + \frac{\frac{1}{4}}{3 !} x^{3} + \frac{- \frac{3}{8}}{4 !} x^{4} + \dots

Refine

= ln (2) - 1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{24} x^{3} - \frac{1}{64} x^{4} + \dots

What is taylor ln(x+2)-1 ?
The answer to taylor ln(x+2)-1 is ln(2)-1+1/2 x-1/8 x^2+1/24 x^3-1/64 x^4+\ldots