Define for and consider the alternating series .

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

Problem

Define cn=2n4−n3n5+7c_{n}=\dfrac{2n^{4}-n}{3n^{5}+7} for n≥1n\ge 1 and consider the alternating series ∑n=1∞(−1)ncn\displaystyle\sum_{n=1}^{\infty}(-1)^{n}c_{n}. (a) Determine whether the series ∑n=1∞∣(−1)ncn∣\displaystyle\sum_{n=1}^{\infty}|(-1)^{n}c_{n}| converges or diverges. Show all supporting work. (b) Classify the alternating series ∑n=1∞(−1)ncn\displaystyle\sum_{n=1}^{\infty}(-1)^{n}c_{n} as absolutely convergent, conditionally convergent, or divergent. (c) Explain in words how the highest-degree terms of the numerator and denominator in cnc_{n} guide your conclusions in parts (a) and (b).

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