# Comparison & limit comparison test

### Comparison & limit comparison test

You may notice that some series look very complicated, but it shares the same properties as another series that looks very simple and easy. In this case, we can use the comparison test or limit comparison test. We will look at what conditions must be met to use these tests, and then use the tests on some complicated looking series. Lastly, we will use both the comparison test and the limit comparison test on a series, and conclude that they give the same result.

#### Lessons

Note *The Comparison test says the following:
Let $\sum a_n$ and $\sum b_n$ be two series where $a_n\leq b_n$ for all $n$ and $a_nb_n\geq0$. Then we say that
1. If $\sum b_n$ is convergent, then $\sum a_n$ is also convergent
2. If $\sum a_n$ is divergent, then $\sum b_n$ is also divergent.

The Limit Comparison Test says the following:
Let $\sum a_n$ and $\sum b_n$ be two series where $a_n\geq 0$ and $b_n$ > 0 for all $n$. Then we say that

$\lim$n →$\infty$ $\frac{a_n}{b_n}=c$

If $c$ is a positive finite number, then either both series converge or diverge.
• Introduction
Overview:

a)
Comparison test

b)
Limit Comparison test

• 1.
Convergence & Divergence of Comparison Tests
Use the Comparison Test to determine if the series converge or diverge.
a)
$\sum_{n=1}^{\infty}\frac{1}{2^n+5}$

b)
$\sum_{n=1}^{\infty}\frac{n^4+5}{n^5-sin^4(2n)}$

c)
$\sum_{n=1}^{\infty}\frac{n^4cos^4(7n)-1}{n^6}$

• 2.
Convergence & Divergence of Limit Comparison Tests
Use the Limit Comparison Test to determine if the series converge or diverge.
a)
$\sum_{n=3}^{\infty}\frac{n^2+n^3}{\sqrt{n^8+n^4}}$

b)
$\sum_{n=1}^{\infty}\frac{1}{n^2-7n-12}$

• 3.
Understanding of Both Tests
Use both the comparison and limit comparison test for the series $\sum_{k=1}^{\infty}\frac{\sqrt{k^3-1}}{k^3-2k^2+5}$. What do both tests say?