Recall that the Maclaurin series for is .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Recall that the Maclaurin series for sin⁡x\sin x is ∑n=0∞(−1)nx2n+1(2n+1)!\displaystyle \sum_{n=0}^{\infty} (-1)^{n}\dfrac{x^{2n+1}}{(2n+1)!}. (1) Write the third-degree and fifth-degree Maclaurin polynomials, S3(x)S_{3}(x) and S5(x)S_{5}(x), for sin⁡x\sin x. (Express each as a finite polynomial.) (2) Compute S3(0.5)S_{3}(0.5) and S5(0.5)S_{5}(0.5), each to eight decimal places. (3) The calculator value of sin⁡(0.5)\sin(0.5) is 0.479425540.47942554. Determine the absolute error of each approximation from part (b). (4) Compare the two errors from part (c) and describe the extent to which increasing the degree improved the accuracy of the approximation.

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