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Chapter 10: Q. 0 (page 777)

Read the section and make your own summary of the material.

Short Answer

Expert verified

The summary of the material is as follows.

Step by step solution

01

Step 1. Definition of Comparison Tests

Analyze series by comparing them with series whose behavior we understand.
Use inequalities to compare series.
Use limits to compare series.

02

Step 2. Summary of the section

Comparison Test:

If ${\sum a_{n}}_{n = 1}^{\infty}$ and ${\sum b_{n}}_{n = 1}^{\infty}$ are series with non-negative terms, then:

If ${\sum b_{n}}_{n = 1}^{\infty}$ converges and $a_{n} \leq b_{n}$ for all $n$ , then ${\sum a_{n}}_{n = 1}^{\infty}$ converges.
If ${\sum b_{n}}_{n = 1}^{\infty}$ diverges and $a_{n} \geq b_{n}$ for all $n$ , then ${\sum a_{n}}_{n = 1}^{\infty}$ diverges.

Limit Comparison Test

Let ${\sum a_{n}}_{n = 1}^{\infty}$ and ${\sum b_{n}}_{n = 1}^{\infty}$ be positive-termed series. If

\underset{n \to \infty}{l i m} \frac{a_{n}}{b_{n}} = c

,

where $c$ is finite, and $c > 0$ , then either both series converge or both diverge.

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