Consider the infinite series S= _ n=1 ^ (-1)^ n n^ 2 3^ n

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

Problem

Consider the infinite series S=∑n=1∞(−1)nn23n.S=\sum_{n=1}^{\infty}(-1)^{n}\frac{n^{2}}{3^{n}}. Answer the following questions. (a) Determine whether the series SS converges or diverges. Show all work leading to your conclusion. (b) Decide whether the convergence found in part (a) is absolute or conditional. Fully justify your decision. (c) State the numerical value of lim⁡n→∞∣(−1)nn23n∣\displaystyle\lim_{n\to\infty}\left|(-1)^{n}\dfrac{n^{2}}{3^{n}}\right| and explain how this limit supports your answer to part (b).

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