泰勒 1/x ,\at 1

解答

泰勒

\frac{1}{x}, at 1

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

求解步骤

应用泰勒公式

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

求导数

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

整理后得

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

\int e^{iπ z} d z

d er i v a t i v e f (t) = t - \frac{1}{t}

\int \frac{sec ( x ) tan ( x )}{sec ( x ) - 2} d x

d er i v a t i v e \frac{2 x ^{3}}{3} + \frac{x ^{2}}{2} + 6 x + 6

x \to 2 lim (ln (x))