Consider the infinite series S= _ n=1 ^ 7n^ 2 -4 5n^ 3 +9

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

Problem

Consider the infinite series S=∑n=1∞7n2−45n3+9.S=\sum_{n=1}^{\infty}\frac{7n^{2}-4}{5n^{3}+9}. Answer the following questions related to the long–term behavior of the terms of SS. (a) Identify the highest–order terms in the numerator and denominator of the general term an=7n2−45n3+9\displaystyle a_n=\frac{7n^{2}-4}{5n^{3}+9} and write a simplified expression that captures their dominant growth rate as n→∞n\to\infty. (b) Using the result from part (a), decide whether the series SS converges or diverges. State your conclusion clearly. (c) Provide a complete justification for your conclusion in part (b), including all necessary limits and inequalities.

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