taylor (-x+ln(1+x))/(x^2)

Lösung

taylor

\frac{- x + ln ( 1 + x )}{x ^{2}}

- 1 + ln (2) + \frac{- 4 ln ( 2 ) + 3}{2} (x - 1) + \frac{24 ln ( 2 ) - 17}{8} (x - 1)^{2} + \frac{- 192 ln ( 2 ) + 134}{48} (x - 1)^{3} + \frac{1920 ln ( 2 ) - 1334}{384} (x - 1)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= - 1 + ln (2) + \frac{\frac{- 4 l n ( 2 ) + 3}{2}}{1 !} (x - 1) + \frac{\frac{24 l n ( 2 ) - 17}{4}}{2 !} (x - 1)^{2} + \frac{\frac{- 192 l n ( 2 ) + 134}{8}}{3 !} (x - 1)^{3} + \frac{\frac{1920 l n ( 2 ) - 1334}{16}}{4 !} (x - 1)^{4} + \dots

Fasse zusammen

= - 1 + ln (2) + \frac{- 4 ln ( 2 ) + 3}{2} (x - 1) + \frac{24 ln ( 2 ) - 17}{8} (x - 1)^{2} + \frac{- 192 ln ( 2 ) + 134}{48} (x - 1)^{3} + \frac{1920 ln ( 2 ) - 1334}{384} (x - 1)^{4} + \dots

\int \frac{1}{x 1 - x} d x

d er i v a t i v e x^{6} (5.1)^{x}

x \to 0 lim ((\frac{1}{x ^{2}})^{x})

\int 7 + 7 tan^{2} (x) d x

\frac{d}{d x} (\frac{3 cos ( x )}{1 - sin ( x )})