泰勒 sqrt(9+x^2)

解答

泰勒

9 + x^{2}

3 + \frac{1}{6} x^{2} - \frac{1}{216} x^{4} + \frac{1}{3888} x^{6} - \frac{5}{279936} x^{8} + \dots

求解步骤

应用泰勒公式

= 3 + \frac{\frac{d}{d x} ( 9 + x ^{2} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( 9 + x ^{2} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 9 + x ^{2} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 3 + \frac{0}{1 !} x + \frac{\frac{1}{3}}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- \frac{1}{9}}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{\frac{5}{27}}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{- \frac{175}{243}}{8 !} x^{8} + \dots

整理后得

= 3 + \frac{1}{6} x^{2} - \frac{1}{216} x^{4} + \frac{1}{3888} x^{6} - \frac{5}{279936} x^{8} + \dots

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