泰勒 log_{10}(x+1)

解答

泰勒

lo g_{10} (x + 1)

\frac{1}{ln ( 10 )} x - \frac{1}{2 ln ( 10 )} x^{2} + \frac{1}{3 ln ( 10 )} x^{3} - \frac{1}{4 ln ( 10 )} x^{4} + \frac{1}{5 ln ( 10 )} x^{5} + \dots

求解步骤

应用泰勒公式

= 0 + \frac{\frac{d}{d x} ( lo g _{10} ( x + 1 ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( lo g _{10} ( x + 1 ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( lo g _{10} ( x + 1 ) ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 0 + \frac{\frac{1}{l n ( 10 )}}{1 !} x + \frac{- \frac{1}{l n ( 10 )}}{2 !} x^{2} + \frac{\frac{2}{l n ( 10 )}}{3 !} x^{3} + \frac{- \frac{6}{l n ( 10 )}}{4 !} x^{4} + \frac{\frac{24}{l n ( 10 )}}{5 !} x^{5} + \dots

整理后得

= \frac{1}{ln ( 10 )} x - \frac{1}{2 ln ( 10 )} x^{2} + \frac{1}{3 ln ( 10 )} x^{3} - \frac{1}{4 ln ( 10 )} x^{4} + \frac{1}{5 ln ( 10 )} x^{5} + \dots

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