taylor ln(x+1+sqrt(x^2+2x)),1

Lösung

taylor

ln (x + 1 + x^{2} + 2 x), 1

ln (2 + 3) + \frac{1}{3} (x - 1) - \frac{1}{3 3} (x - 1)^{2} + \frac{1}{6 \cdot 3 ^{\frac{1}{2}}} (x - 1)^{3} - \frac{11}{108 3} (x - 1)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= ln (2 + 3) + \frac{\frac{1}{3}}{1 !} (x - 1) + \frac{- \frac{2}{3 3}}{2 !} (x - 1)^{2} + \frac{\frac{1}{3 ^{\frac{1}{2}}}}{3 !} (x - 1)^{3} + \frac{- \frac{22}{9 3}}{4 !} (x - 1)^{4} + \dots

Fasse zusammen

= ln (2 + 3) + \frac{1}{3} (x - 1) - \frac{1}{3 3} (x - 1)^{2} + \frac{1}{6 \cdot 3 ^{\frac{1}{2}}} (x - 1)^{3} - \frac{11}{108 3} (x - 1)^{4} + \dots