For a real parameter p>0, consider the family of series
T(p)=n=2∑∞n(lnn)p1.
Answer the following questions.
a. Determine whether T(1) converges or diverges. Provide a complete justification.
b. Determine for which values of p the series T(p) converges absolutely. Include justification for any comparison or limiting argument you use.
c. Determine for which values of p the series T(p) diverges. Explain why no conditional convergence occurs for this family.
d. Summarize your findings by describing how the behavior of T(p) changes as p varies, and interpret why the value p=1 represents a critical threshold for convergence.