taylor x/(e^x-1)

Lösung

taylor

\frac{x}{e ^{x} - 1}

\frac{1}{e - 1} - \frac{1}{( e - 1 ) ^{2}} (x - 1) - \frac{e ^{2} - 3 e}{2 ( e - 1 ) ^{3}} (x - 1)^{2} + \frac{2 e ^{3} - 4 e ^{2} - 4 e}{6 ( e - 1 ) ^{4}} (x - 1)^{3} - \frac{3 e ^{4} + e ^{3} - 23 e ^{2} - 5 e}{24 ( e - 1 ) ^{5}} (x - 1)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{1}{e - 1} + \frac{- \frac{1}{( e - 1 ) ^{2}}}{1 !} (x - 1) + \frac{- \frac{e ^{2} - 3 e}{( e - 1 ) ^{3}}}{2 !} (x - 1)^{2} + \frac{\frac{2 e ^{3} - 4 e ^{2} - 4 e}{( e - 1 ) ^{4}}}{3 !} (x - 1)^{3} + \frac{- \frac{3 e ^{4} + e ^{3} - 23 e ^{2} - 5 e}{( e - 1 ) ^{5}}}{4 !} (x - 1)^{4} + \dots

Fasse zusammen

= \frac{1}{e - 1} - \frac{1}{( e - 1 ) ^{2}} (x - 1) - \frac{e ^{2} - 3 e}{2 ( e - 1 ) ^{3}} (x - 1)^{2} + \frac{2 e ^{3} - 4 e ^{2} - 4 e}{6 ( e - 1 ) ^{4}} (x - 1)^{3} - \frac{3 e ^{4} + e ^{3} - 23 e ^{2} - 5 e}{24 ( e - 1 ) ^{5}} (x - 1)^{4} + \dots

\frac{d}{d x} (2 ln (x^{2}))

n = 1 \sum \infty 1 0^{n} x^{n}

\int (x^{2} + 5 x + 6) cos (2 x) d x

a re a y = x, g (y, x) = x^{2}, [0, 1]

\frac{d}{d x} (cos (sin^{3} (5 x - 5)))