taylor e^{(-x^2)/2}

Lösung

taylor

e^{\frac{- x ^{2}}{2}}

1 - \frac{1}{2} x^{2} + \frac{1}{8} x^{4} - \frac{1}{48} x^{6} + \frac{1}{384} x^{8} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} x + \frac{- 1}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{3}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 15}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{105}{8 !} x^{8} + \dots

Fasse zusammen

= 1 - \frac{1}{2} x^{2} + \frac{1}{8} x^{4} - \frac{1}{48} x^{6} + \frac{1}{384} x^{8} + \dots