Let and be real numbers and define .

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

Problem

Let pp and qq be real numbers and define T(p,q)=∑n=1∞npnq+1T(p,q)=\displaystyle \sum_{n=1}^{\infty} \dfrac{n^{p}}{n^{q}+1}. (a) Show that L(p,q)=lim⁡n→∞npnq+1L(p,q)=\displaystyle \lim_{n\to\infty} \dfrac{n^{p}}{n^{q}+1} equals 00 if p<qp<q, equals 11\dfrac{1}{1} if p=qp=q, and is unbounded if p>qp>q. Provide complete justification for each case. (b) Using the result of part (a), determine precisely the set of ordered pairs (p,q)(p,q) for which the series T(p,q)T(p,q) cannot converge. Describe this set as a region in the pqpq–plane and defend your conclusion with careful reasoning. (c) Treating the boundary case p=qp=q, evaluate the first four non-zero terms explicitly when p=q=1p=q=1 and interpret whether the limit information alone decides convergence. Explain why or why not. (d) For p<qp<q, explain why the limit you obtained in part (a) supplies no information about convergence. State (without carrying out) two different types of analyses that could be used to settle convergence in that region.

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