maclaurin 1-e^{-x}

Lösung

maclaurin

1 - e^{- x}

x - \frac{1}{2} x^{2} + \frac{1}{6} x^{3} - \frac{1}{24} x^{4} + \frac{1}{120} x^{5} + \dots

Schritte zur Lösung

Apply the Maclaurin Formula

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

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{- 1}{2 !} x^{2} + \frac{1}{3 !} x^{3} + \frac{- 1}{4 !} x^{4} + \frac{1}{5 !} x^{5} + \dots

Fasse zusammen

= x - \frac{1}{2} x^{2} + \frac{1}{6} x^{3} - \frac{1}{24} x^{4} + \frac{1}{120} x^{5} + \dots

d er i v a t i v e \frac{x ^{2}}{4 + x}

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x \to 1 lim ([2 x])

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\frac{d ^{2} S}{d x ^{2}} = x