1. Lesson 11-4
Comparison Tests

2. Another Series Type
p-Series – summation of the following infinite sequence ∑ is convergent if p > 1 and divergent if p ≤ 1 1 np

3. Types of Series
Geometric
Telescoping
Harmonic
P-Series

4. Comparison Test
Suppose the series and are series with positive terms. 1) if is convergent and an ≤ bn for all n, then is also convergent 2) if is divergent and an ≤ bn for all n, then is also divergent
∑ an
i=1 ∞
∑ bn
i=1 ∞
∑ bn
i=1 ∞
∑ an
i=1 ∞
∑ bn
i=1 ∞
∑ an
i=1 ∞

5. Limit Comparison Test
Suppose the series and are series with positive terms. If where c is a finite number and c > 0, then either both series converge or both diverge.
∑ an
i=1 ∞
∑ bn
i=1 ∞ an
Lim = c bn
n→∞

6. 11-4 Example 1 Is the following series convergent or divergent? 2
Lim an = 0 so it might converge
n→∞
p Series: with p = 0.85
so it will diverge since p < 1 6

7. 11-4 Example 2 Is the following series convergent or divergent? series
Lim = so it might converge
n→∞
n² +1
< (p Series: with p = 2) n² n²
so by the comparison test it is convergent 7

8. 11-4 Example 3
Examine the series below and determine if it converges or diverges.
cos² n
series nth term is
n² + 1
Lim an = 0 therefore the series might converge
n→∞
cos² n
Cos² n is ≤ 1 so <
n² n²
Since 1/n² converges, therefore the series converges 8

9. Homework
Pg 734 – 735: problems 7, 10, 11, 18, 25