For a real parameter

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

Problem

For a real parameter p>0p>0, consider the family of series T(p)=∑n=2∞1n(ln⁡n)p.T(p)=\sum_{n=2}^{\infty}\dfrac{1}{n(\ln n)^{p}}. Answer the following questions. (1) Determine whether T(1)T(1) converges or diverges. Provide a complete justification. (2) Determine for which values of pp the series T(p)T(p) converges absolutely. Include justification for any comparison or limiting argument you use. (3) Determine for which values of pp the series T(p)T(p) diverges. Explain why no conditional convergence occurs for this family. (4) Summarize your findings by describing how the behavior of T(p)T(p) changes as pp varies, and interpret why the value p=1p=1 represents a critical threshold for convergence.

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