Consider the Maclaurin series of

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

Problem

Consider the Maclaurin series of sin⁡x\sin x, sin⁡x=x−x33!+x55!−x77!+⋯ .\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!}+\cdots. The goal is to approximate sin⁡(1)\sin(1) to within 10−610^{-6}. (a) Compute the degree-5 polynomial obtained by truncating the series after the x5x^{5} term. Evaluate this polynomial at x=1x=1. (b) Determine whether the degree-5 approximation already meets the desired accuracy. Justify your conclusion with an explicit numerical comparison. (c) If additional terms are required, find the minimal degree nn that guarantees the absolute error is below 10−610^{-6}, then state the corresponding approximation to eight decimal places.

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