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

Lösung

taylor

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

1 - x^{2} + x^{4} - x^{6} + x^{8} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} x + \frac{- 2}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{24}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 720}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{40320}{8 !} x^{8} + \dots

Fasse zusammen

= 1 - x^{2} + x^{4} - x^{6} + x^{8} + \dots

\frac{d y}{d x} = 8 x^{3} e^{- 2 y}

\frac{d}{d t} (t e^{5 t})

\int (x^{2} - 1) d x

\int 2 t \cdot e^{t} d t

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