泰勒 sin(pix), 1/2

解答

泰勒

sin (π x), \frac{1}{2}

1 - \frac{π ^{2}}{2} (x - \frac{1}{2})^{2} + \frac{π ^{4}}{24} (x - \frac{1}{2})^{4} - \frac{π ^{6}}{720} (x - \frac{1}{2})^{6} + \frac{π ^{8}}{40320} (x - \frac{1}{2})^{8} + \dots

求解步骤

应用泰勒公式

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

求导数

= 1 + \frac{0}{1 !} (x - \frac{1}{2}) + \frac{- π ^{2}}{2 !} (x - \frac{1}{2})^{2} + \frac{0}{3 !} (x - \frac{1}{2})^{3} + \frac{π ^{4}}{4 !} (x - \frac{1}{2})^{4} + \frac{0}{5 !} (x - \frac{1}{2})^{5} + \frac{- π ^{6}}{6 !} (x - \frac{1}{2})^{6} + \frac{0}{7 !} (x - \frac{1}{2})^{7} + \frac{π ^{8}}{8 !} (x - \frac{1}{2})^{8} + \dots

整理后得

= 1 - \frac{π ^{2}}{2} (x - \frac{1}{2})^{2} + \frac{π ^{4}}{24} (x - \frac{1}{2})^{4} - \frac{π ^{6}}{720} (x - \frac{1}{2})^{6} + \frac{π ^{8}}{40320} (x - \frac{1}{2})^{8} + \dots

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