maclaurin sin(pix)

Lösung

maclaurin

sin (π x)

π x - \frac{π ^{3}}{6} x^{3} + \frac{π ^{5}}{120} x^{5} - \frac{π ^{7}}{5040} x^{7} + \frac{π ^{9}}{362880} x^{9} + \dots

Schritte zur Lösung

Apply the Maclaurin Formula

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

Evaluate Derivatives

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

Fasse zusammen

= π x - \frac{π ^{3}}{6} x^{3} + \frac{π ^{5}}{120} x^{5} - \frac{π ^{7}}{5040} x^{7} + \frac{π ^{9}}{362880} x^{9} + \dots

t \to 0 lim (\frac{7}{1 + t})

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\frac{d}{d x} (\frac{1 - 3 x}{4 + x})

d er i v a t i v e y = 9 x^{2} e^{x}

n = 1 \sum \infty (- 5)^{- n}