Consider the infinite series S(p)= _ n=1 ^ 1 n^ p (p R )
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the infinite series S(p)=n=1∑∞np1(p∈R).
This classic one–parameter family provides a benchmark for comparing more complicated series.
(a) Determine all real numbers p for which the sequence of terms an=np1 approaches 0 as n→∞. Justify your answer.
(b) For which values of p does the series S(p) converge? For which values does it diverge? Provide a complete explanation that references the behavior of an appropriate improper integral without actually evaluating that integral explicitly.
(c) Describe, in clear prose, how the critical value of p you found in part (b) serves as a threshold separating two qualitatively different long–run behaviors for S(p).