Two students are discussing why some infinite series of the form converge while…

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

Problem

Two students are discussing why some infinite series of the form ∑n=1∞1np\displaystyle\sum_{n=1}^{\infty}\frac1{n^{p}} converge while others do not. Student A claims, “The series with p=1p=1 should converge because its terms approach 00.” Student B responds, “Actually, the series with p=1p=1 diverges, but the series with p=2p=2 converges.” Address the points raised by the two students and reconcile their disagreement. (a) Describe—in words and/or symbolic reasoning—why the fact that lim⁡n→∞1np=0\displaystyle\lim_{n\to\infty}\frac1{n^{p}}=0 is necessary but not sufficient for ∑n=1∞1np\displaystyle\sum_{n=1}^{\infty}\frac1{n^{p}} to converge. (b) Provide a logical argument demonstrating that the harmonic series ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty}\frac1n diverges. (c) Provide a logical argument demonstrating that the series ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty}\frac1{n^{2}} converges. (d) Explain how your arguments in parts (b) and (c) together clarify which values of pp guarantee convergence for a pp-series.

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