泰勒 sqrt(x),x=4

解答

泰勒

x, x = 4

2 + \frac{1}{4} (x - 4) - \frac{1}{64} (x - 4)^{2} + \frac{1}{512} (x - 4)^{3} - \frac{5}{16384} (x - 4)^{4} + \dots

求解步骤

应用泰勒公式

= 2 + \frac{\frac{d}{d x} ( x ) ( 4 )}{1 !} (x - 4) + \frac{\frac{d ^{2}}{d x ^{2}} ( x ) ( 4 )}{2 !} (x - 4)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x ) ( 4 )}{3 !} (x - 4)^{3} + \dots

求导数

= 2 + \frac{\frac{1}{4}}{1 !} (x - 4) + \frac{- \frac{1}{32}}{2 !} (x - 4)^{2} + \frac{\frac{3}{256}}{3 !} (x - 4)^{3} + \frac{- \frac{15}{2048}}{4 !} (x - 4)^{4} + \dots

整理后得

= 2 + \frac{1}{4} (x - 4) - \frac{1}{64} (x - 4)^{2} + \frac{1}{512} (x - 4)^{3} - \frac{5}{16384} (x - 4)^{4} + \dots