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MathCalculus

Identify the Function Represented by the Following Power Series

Grouping each repeating sign block turns the three series into geometric sums: (x+x²−x³)/(1−x³), (x+x²−x³−x⁴)/(1−x⁴) = x(1+x)/(1+x²), and (x−x²−x³)/(1−x³), all for |x| < 1.

Question

In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function.

  1. x+x2−x3+x4+x5−x6+⋯x + x^{2} - x^{3} + x^{4} + x^{5} - x^{6} + \cdots (Hint: Group powers x3kx^{3k}, x3k−1x^{3k-1} and x3k−2x^{3k-2}.)

  2. x+x2−x3−x4+x5+x6−x7−x8+⋯x + x^{2} - x^{3} - x^{4} + x^{5} + x^{6} - x^{7} - x^{8} + \cdots (Hint: Group powers x4kx^{4k}, x4k−1x^{4k-1}, etc.)

  3. x−x2−x3+x4−x5−x6+x7−⋯x - x^{2} - x^{3} + x^{4} - x^{5} - x^{6} + x^{7} - \cdots (Hint: Group powers x3kx^{3k}, x3k−1x^{3k-1} and x3k−2x^{3k-2}.)

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Evidence boundary

This page is a normalized presentation of the OpenStax Calculus Volume 2 section 6.2 exercise set that asks for the sum of a power series in terms of geometric series and then as a rational function. Items 79 to 81 are reproduced as published, including the sign of every listed term and the grouping hints; item 82 of the same set is not used because its first term is rendered inconsistently across sources. Each answer is stated with the interval on which the geometric ratio satisfies |r| < 1, and the rational function is claimed to equal the series only inside that interval.