Let (T= _ k=1 ^ (-1)^ k

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

Problem

Let T=∑k=1∞(−1)k kekT=\displaystyle\sum_{k=1}^{\infty}(-1)^{k}\,\dfrac{k}{e^{k}}. Student C truncates the series after the third term, reporting P3P_3. Student D truncates after the fifth term, reporting P5P_5. Each student records all calculations to eight decimal places. (a) Show that the terms ak=(−1)k keka_k=(-1)^k\,\dfrac{k}{e^{k}} satisfy the necessary conditions for convergence of an alternating series. (b) Establish an upper bound for ∣T−P3∣|T-P_3|. (c) Establish an upper bound for ∣T−P5∣|T-P_5|. (d) Suppose a hand-held calculator requires 0.04 seconds to evaluate each individual term. Using your results from parts (b) and (c), determine which student attains the smaller maximum possible error per second of computation. Justify your answer quantitatively.

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