maclaurin ln(cos(x))

Lösung

maclaurin

ln (cos (x))

- \frac{1}{2} x^{2} - \frac{1}{12} x^{4} - \frac{1}{45} x^{6} - \frac{17}{2520} x^{8} - \frac{31}{14175} x^{10} + \dots

Schritte zur Lösung

Apply the Maclaurin Formula

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

Evaluate Derivatives

= 0 + \frac{0}{1 !} x + \frac{- 1}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- 2}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 16}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{- 272}{8 !} x^{8} + \frac{0}{9 !} x^{9} + \frac{- 7936}{10 !} x^{10} + \dots

Fasse zusammen

= - \frac{1}{2} x^{2} - \frac{1}{12} x^{4} - \frac{1}{45} x^{6} - \frac{17}{2520} x^{8} - \frac{31}{14175} x^{10} + \dots

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

d er i v a t i v e (7 x^{2} - 12 x) e^{x}

\frac{d}{d x} (x^{3} (4 - x))

n = 0 \sum \infty \frac{4}{n ( n + 2 )}

n = 0 \sum \infty \frac{( 5 ^{n} )}{n !}