A mathematical model generates three infinite series whose sums influence different…

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

Problem

A mathematical model generates three infinite series whose sums influence different parameters in the model. T1=∑n=1∞5n+43n+1T_1=\displaystyle \sum_{n=1}^{\infty} \frac{5n+4}{3n+1} T2=∑n=1∞(−2)n1nT_2=\displaystyle \sum_{n=1}^{\infty} (-2)^{n}\frac{1}{n} T3=∑n=2∞1n(ln⁡n)2T_3=\displaystyle \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}} (a) For each series, find lim⁡n→∞an\displaystyle \lim_{n \to \infty} a_n, where ana_n is the nnth term. (b) Using only the information from part (a), identify every series that must diverge. Clearly state your justification for each decision. (c) For any series not classified as divergent in part (b), discuss whether the limit test is inconclusive and what, if anything, can be concluded about its convergence on the basis of the test.

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