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Popular Calculus >

taylor ln(x),\at 1

Solution

taylor

ln (x), at 1

Solution

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

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

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

Refine

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

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