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

x \to - 5 lim (\frac{x ^{2} - 6}{5 - x})

t an g e n t y = x + \frac{3}{x}, (3, 4)

\frac{\partial}{\partial y} (- x y sin (x) + 2 y cos (x))

\frac{d y}{d x} = \frac{( 1 + y )}{x}

s l o p e (0.2) (1.5)