Calculus 8th Edition

Published by Cengage

Chapter 11 - Infinite Sequences and Series - 11.5 Alternating Series - 11.5 Exercises - Page 776: 33

Answer

$p$ is not a negative integer.

Work Step by Step

1. We will have to apply the Test of Divergence to find the value of $p$ for the given alternating series when $p\geq 0$ So, $\lim\limits_{n \to \infty}\dfrac{1}{n+p}=0$ This implies that the limit $0$ satisfies all the condition for alternating series test and converge by the Test of Divergence. 2. We will have to apply the Test of Divergence to find the value of $p$ for the given alternating series when $p\geq 0$ $p \lt 0$ Here, $\lim\limits_{n \to \infty}\dfrac{1}{n+p}$ the limit is undefined because $n=-p$ , that is, the series will not converge by the Test of Divergence. But the denominator will not become $0$ for the negative values. This implies that $p$ can be any real value but the value of $p$ can not be a negative integer.

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