Four infinite series are defined below

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

Problem

Four infinite series are defined below. S₁: ∑n=2∞ln⁡nn3/2\displaystyle\sum_{n=2}^{\infty}\frac{\ln n}{n^{3/2}} S₂: ∑n=1∞n!(2n)!\displaystyle\sum_{n=1}^{\infty}\frac{n!}{(2n)!} S₃: ∑n=1∞(3n+13n−2)n\displaystyle\sum_{n=1}^{\infty}\left(\frac{3n+1}{3n-2}\right)^{n} S₄: ∑n=3∞1n(ln⁡n)2\displaystyle\sum_{n=3}^{\infty}\frac{1}{n(\ln n)^2} For each series answer the questions in Parts (a) and (b). (1) Indicate a convergence test that would be most efficient for analyzing each series. Provide a brief justification for the suitability of the chosen test in each case. (2) Determine whether each series converges or diverges. Supply clear reasoning that supports your conclusion, citing any necessary conditions required by the test you employ.

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