Determine if the p-series converges or diverges. n = 1 1 n 22

Question:

Determine if the p-series converges or diverges.

{eq}\displaystyle \sum_{n=1}^\infty \frac{1}{n^{22}}{/eq}

p-Series:

A p-series is a type of infinite series. A p-series can be identified as a series of fractions that have the same numerator, and all succeeding denominator terms are the index variable raised to a constant power. That is,

{eq}\displaystyle \sum_{n = 1}^\infty \frac{1}{n^p} = \frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3} + ... {/eq}

A p-series is convergent as long as {eq}\displaystyle p > 1 {/eq}.

Answer and Explanation: 1

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We are given:

{eq}\displaystyle \sum_{n = 1}^\infty \frac{1}{n^{22}} {/eq}

This is a standard p-series with {eq}\displaystyle p = 22 {/eq}. The...

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P-Series: Definition & Examples

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Chapter 29 / Lesson 5
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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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