maclaurin ln(2+x)

Lösung

maclaurin

ln (2 + x)

ln (2) + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{24} x^{3} - \frac{1}{64} x^{4} + \dots

Schritte zur Lösung

Apply the Maclaurin Formula

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

Evaluate Derivatives

= ln (2) + \frac{\frac{1}{2}}{1 !} x + \frac{- \frac{1}{4}}{2 !} x^{2} + \frac{\frac{1}{4}}{3 !} x^{3} + \frac{- \frac{3}{8}}{4 !} x^{4} + \dots

Fasse zusammen

= ln (2) + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{24} x^{3} - \frac{1}{64} x^{4} + \dots

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\frac{1}{2} y^{^{''}} + 2 y = tan (2 t) - \frac{1}{2} e^{t}