An electronic sensor reports a rapidly changing voltage once every millisecond

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

Problem

An electronic sensor reports a rapidly changing voltage V(t)V(t) once every millisecond. To compensate for a systematic offset, engineers alternate between adding and subtracting a small calibration factor whose magnitude changes with the sample index nn according to cn=0.02nc_{n}=\dfrac{0.02}{\sqrt n} volts. Over time they will apply the cumulative correction C=∑n=1∞(−1)n cn=∑n=1∞(−1)n 0.02n.C=\sum_{n=1}^{\infty}(-1)^{n}\,c_{n}=\sum_{n=1}^{\infty}(-1)^{n}\,\frac{0.02}{\sqrt n}. Answer the following questions to assess whether the total correction CC is finite. (a) Compute lim⁡n→∞0.02n\displaystyle\lim_{n\to\infty}\dfrac{0.02}{\sqrt n} and interpret its significance in the present context. (b) Determine whether the sequence {0.02n}\bigl\{\dfrac{0.02}{\sqrt n}\bigr\} is eventually non-increasing and explain briefly. (c) Based on your work in parts (a) and (b), decide whether the infinite sum CC converges or diverges. Provide a clear justification that references the outcome of parts (a) and (b). (d) Suppose instead the engineers choose cn=0.02n1.1c_{n}=\dfrac{0.02}{n^{1.1}}. Without re-deriving each earlier step, explain how this change affects the convergence conclusion and why.

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