For the following statement, answer "true" if both the argument and conclusion are valid....

Question:

For the following statement, answer "true" if both the argument and conclusion are valid. Otherwise, answer "false".

For all {eq}n > 2 {/eq}, {eq}\displaystyle \frac{n}{n^3 - 5} < \frac{2}{n^2} {/eq}, and the series {eq}\displaystyle 2 \sum \frac{1}{n^2} {/eq} converges, so by the Comparison Test, the series {eq}\displaystyle \sum \frac{n}{n^3 - 5} {/eq} converges.

P-series Test

The p-series test for convergence of series states that if we have a series in the form {eq}\sum \dfrac{1}{n^p} {/eq}, then that series is convergent if the value of {eq}p {/eq} is greater than 1. Otherwise, the series is divergent.

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First, observe that

{eq}\begin{align*} & \dfrac{n}{n^3-5} < \dfrac{2}{n^2} \\ \iff & n^3 < 2n^3 - 10 \\ \iff & 10 <...

P-Series: Definition & Examples

from

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.