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Use the P-Test to determine whether the series converges or diverges: a. 1 n 8 n=1

Question:

Use the P-Test to determine whether the series converges or diverges:

a. {eq}\sum \infty \frac{1}{n^{8}} {/eq}

n=1

P-Series

If we have a series in the form

{eq}\sum^{\infty}_{n=1} \dfrac{1}{n^p} = \dfrac{1}{1^p} + \dfrac{1}{2^p} + \dfrac{1}{3^p} + \cdots {/eq}

then this series is called a {eq}p {/eq}-series. Now, the P-test states that a {eq}p {/eq}-series is convergent if the value of {eq}p {/eq} is greater than 1. Otherwise, the {eq}p {/eq}-series is divergent.

Answer and Explanation: 1

Consider the series

{eq}\sum^{\infty}_{n=1} \dfrac{1}{n^8} {/eq}

Note that the above series is a {eq}p {/eq}-series with {eq}p=8 {/eq}. Now, since the value of {eq}p {/eq} is greater than 1, by the P-test, the series is convergent.


Learn more about this topic:

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P-Series: Definition & Examples

from

Chapter 29 / Lesson 5
23K

This lesson is designed to help you understand a specific type of series called a p-series. You will determine if a series is a p-series, and you will learn to decide if a p-series converges or diverges.


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