Q. 0 - Calculus

Question

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

Step-by-Step Solution

Answer

The summary of the material is as follows.

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. 

Step 2. Summary of the section

Comparison Test:

If $\underset{n=1}{\overset{\infty }{\sum a_n}}$ and $\underset{n=1}{\overset{\infty }{\sum b_n}}$are series with non-negative terms, then:

1.  If $\underset{n=1}{\overset{\infty }{\sum b_n}}$ converges and  $a_n\le b_n$ for all $n$, then $\underset{n=1}{\overset{\infty }{\sum a_n}}$ converges.
2. If $\underset{n=1}{\overset{\infty }{\sum b_n}}$ diverges and $a_n\ge b_n$ for all $n$, then $\underset{n=1}{\overset{\infty }{\sum a_n}}$diverges.

Limit Comparison Test

Let $\underset{n=1}{\overset{\infty }{\sum a_n}}$ and $\underset{n=1}{\overset{\infty }{\sum b_n}}$ be positive-termed series.  If

$\underset{n\to \infty }{lim}\frac{a_n}{b_n}=c$,

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