題目 8. (10 points) Write down the Taylor series expansion for the function f(x)=x31+x2f(x)=\dfrac{x^3}{1+x^2}f(x)=1+x2x3 at x=0x=0x=0. 解答 解法一 x31+x2=x3∑n=0∞(−x2)n=∑n=0∞(−1)nx2n+3,∣x∣<1.\frac{x^3}{1+x^2}=x^3\sum_{n=0}^\infty(-x^2)^n =\boxed{\sum_{n=0}^\infty(-1)^nx^{2n+3}},\qquad |x|<1.1+x2x3=x3n=0∑∞(−x2)n=n=0∑∞(−1)nx2n+3,∣x∣<1.
