In the study of infinite series

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

Problem

In the study of infinite series, insight can often be gained by examining the long–run behavior of the individual terms themselves. Let {bn}\{b_n\} be a sequence of real numbers and consider the series ∑n=1∞bn\displaystyle \sum_{n=1}^{\infty} b_n. (a) Give a logical argument showing that if ∑n=1∞bn\displaystyle \sum_{n=1}^{\infty} b_n converges, then lim⁡n→∞bn=0\displaystyle \lim_{n\to\infty} b_n = 0. Your explanation should rely on properties of convergent series rather than any external convergence tests. (b) Compute lim⁡n→∞1n+2\displaystyle \lim_{n\to\infty} \frac{1}{n+2}. Explain whether the series ∑n=1∞1n+2\displaystyle \sum_{n=1}^{\infty} \frac{1}{n+2} satisfies the condition you established in part (a). (c) Based on your work in part (b), determine whether ∑n=1∞1n+2\displaystyle \sum_{n=1}^{\infty} \frac{1}{n+2} converges or diverges. Provide a clear justification for your conclusion.

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