Lesson 11-4 Comparison Tests

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

Types of Series Geometric Telescoping Harmonic P-Series

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 ∞

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→∞

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

11-4 Example 2 Is the following series convergent or divergent? series Lim = 0 so it might converge n→∞ 1 -------- n² +1 1 1 -------- < ------- (p Series: with p = 2) n² +1 n² so by the comparison test it is convergent 7

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 1 Cos² n is ≤ 1 so ---------- < ------- n² + 1 n² Since 1/n² converges, therefore the series converges 8

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