What is taylor ln(x),8 ?

The answer to taylor ln(x),8 is 3ln(2)+1/8 (x-8)-1/128 (x-8)^2+1/1536 (x-8)^3-1/16384 (x-8)^4+\ldots

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taylor ln(x),8

taylor

ln (x), 8

3 ln (2) + \frac{1}{8} (x - 8) - \frac{1}{128} (x - 8)^{2} + \frac{1}{1536} (x - 8)^{3} - \frac{1}{16384} (x - 8)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 3 ln (2) + \frac{\frac{1}{8}}{1 !} (x - 8) + \frac{- \frac{1}{64}}{2 !} (x - 8)^{2} + \frac{\frac{1}{256}}{3 !} (x - 8)^{3} + \frac{- \frac{3}{2048}}{4 !} (x - 8)^{4} + \dots

Refine

= 3 ln (2) + \frac{1}{8} (x - 8) - \frac{1}{128} (x - 8)^{2} + \frac{1}{1536} (x - 8)^{3} - \frac{1}{16384} (x - 8)^{4} + \dots

What is taylor ln(x),8 ?
The answer to taylor ln(x),8 is 3ln(2)+1/8 (x-8)-1/128 (x-8)^2+1/1536 (x-8)^3-1/16384 (x-8)^4+\ldots