The Maclaurin series for is

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The Maclaurin series for e−1e^{-1} is 1−1+12!−13!+14!−…1-1+\dfrac{1}{2!}-\dfrac{1}{3!}+\dfrac{1}{4!}-\dots. Let S4S_4 be the partial sum through the 14!\dfrac{1}{4!} term, and let M4=S4+S52M_4 = \dfrac{S_4+S_5}{2} be the average of the 4-term and 5-term partial sums. Compute the maximum possible error for each approximation and state which has the smaller bound.

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