taylor (x+1)^{1/2}

Lösung

taylor

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

1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{16} x^{3} - \frac{5}{128} x^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{\frac{1}{2}}{1 !} x + \frac{- \frac{1}{4}}{2 !} x^{2} + \frac{\frac{3}{8}}{3 !} x^{3} + \frac{- \frac{15}{16}}{4 !} x^{4} + \dots

Fasse zusammen

= 1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{16} x^{3} - \frac{5}{128} x^{4} + \dots

y^{^{'}} = 6 x (y - 1)^{\frac{2}{3}}

y^{^{''}} + 214 y^{^{'}} + 14 y = 0

y^{^{'}} + \frac{3}{x} y = x^{2} y^{2}

\frac{\partial}{\partial x} (e^{3 x} cos (2 x))

\int \frac{1}{x ^{4}} d x