taylor 1/(x+5)

Lösung

taylor

\frac{1}{x + 5}

\frac{1}{5} - \frac{1}{25} x + \frac{1}{125} x^{2} - \frac{1}{625} x^{3} + \frac{1}{3125} x^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

= \frac{1}{5} + \frac{\frac{d}{d x} ( \frac{1}{x + 5} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{x + 5} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{x + 5} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= \frac{1}{5} + \frac{- \frac{1}{25}}{1 !} x + \frac{\frac{2}{125}}{2 !} x^{2} + \frac{- \frac{6}{625}}{3 !} x^{3} + \frac{\frac{24}{3125}}{4 !} x^{4} + \dots

Fasse zusammen

= \frac{1}{5} - \frac{1}{25} x + \frac{1}{125} x^{2} - \frac{1}{625} x^{3} + \frac{1}{3125} x^{4} + \dots