Let be the infinite series where a_n= 3^ n

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

Problem

Let SS be the infinite series S=∑n=1∞an\displaystyle S=\sum_{n=1}^{\infty} a_n where an=3n n!nn.a_n=\frac{3^{n}\, n!}{n^{n}}. Answer the following. (a) Derive a simplified expression for the ratio an+1an\dfrac{a_{n+1}}{a_n} in terms of nn. Do not evaluate any limits in this part. (b) Evaluate lim⁡n→∞an+1an\displaystyle \lim_{n\to\infty}\dfrac{a_{n+1}}{a_n}. Show all steps in your limit calculation and justify each transformation. (c) Determine whether the series SS converges or diverges. Provide a complete justification based on your result in part (b). Clearly state whether your conclusion concerns absolute convergence, conditional convergence, or divergence.

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