Let an=n2+1n3−n(n≥2),S=∑n=2∞an.a_n=\frac{n^{2}+1}{n^{3}-n}\qquad (n\ge 2),\qquad \quad S=\sum_{n=2}^{\infty}a_n.…

Mathematics

Problem

Let an=n2+1n3−n(n≥2),S=∑n=2∞an.a_n=\frac{n^{2}+1}{n^{3}-n}\qquad (n\ge 2),\qquad \quad S=\sum_{n=2}^{\infty}a_n. Assume standard facts about pp-series. (a) Choose an elementary benchmark series whose general term is of the form c/npc/n^{p} (c>0c>0) that will allow you to decide whether SS converges or diverges. Explain why this choice is appropriate. (b) Compute lim⁡n→∞anbn\displaystyle \lim_{n\to\infty}\frac{a_n}{b_n}, where bnb_n is the general term of your benchmark from part (a). Show all work leading to the limit. (c) Determine whether the series SS converges or diverges. Justify your conclusion using the result of part (b) and the known behavior of the benchmark series.

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