テイラー (log_{10}(2-x))/(x+3)

解

テイラー

\frac{lo g _{10} ( 2 - x )}{x + 3}

\frac{lo g _{10} ( 2 )}{3} + \frac{- 2 ln ( 2 ) - 3}{18 ln ( 10 )} x + \frac{24 ln ( 2 ) + 9}{648 ln ( 10 )} x^{2} + \frac{- 288 ln ( 2 ) - 432}{23328 ln ( 10 )} x^{3} + \frac{1492992 ln ( 2 ) + 349920}{362797056 ln ( 10 )} x^{4} + \dots

解答ステップ

テイラーの公式を適用する

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

導関数を評価する

= \frac{lo g _{10} ( 2 )}{3} + \frac{\frac{- 2 l n ( 2 ) - 3}{18 l n ( 10 )}}{1 !} x + \frac{\frac{24 l n ( 2 ) + 9}{324 l n ( 10 )}}{2 !} x^{2} + \frac{\frac{- 288 l n ( 2 ) - 432}{3888 l n ( 10 )}}{3 !} x^{3} + \frac{\frac{1492992 l n ( 2 ) + 349920}{15116544 l n ( 10 )}}{4 !} x^{4} + \dots

改良

= \frac{lo g _{10} ( 2 )}{3} + \frac{- 2 ln ( 2 ) - 3}{18 ln ( 10 )} x + \frac{24 ln ( 2 ) + 9}{648 ln ( 10 )} x^{2} + \frac{- 288 ln ( 2 ) - 432}{23328 ln ( 10 )} x^{3} + \frac{1492992 ln ( 2 ) + 349920}{362797056 ln ( 10 )} x^{4} + \dots