Understanding Integral and Comparison Tests for Series
School
Qatar University**We aren't endorsed by this school
Course
MATH 102
Subject
Mathematics
Date
Dec 10, 2024
Pages
14
Uploaded by ElderMorningSwan51
|M a t h 1 0 2 | S e c 8 . 3| Page1 Section 8.3: The Integral Tests and Comparison Test •The Integral Test •The 𝑝𝑝-series •Comparison Test •Limit Comparison Test After completing this section, you should be able to do the following. •Use the integral test and knowledge of the 𝑝𝑝-series to determine the convergence of a series. •Know that the harmonic series diverges. •Use the comparison test to determine whether a series converges. •Use the limit comparison test to determine whether a series converges
|M a t h 1 0 2 | S e c 8 . 3| Page2 Notation.The notation ∑𝑎𝑎𝑘𝑘without initial and final value of 𝑘𝑘, is used to refer to a general infinite series. The Integral Test: Let ∑𝑎𝑎𝑘𝑘∞𝑘𝑘=𝑙𝑙be an infinite series with positive terms. Let that 𝑓𝑓(𝑥𝑥)be a continuous, positive, decreasing function, for 𝒙𝒙 ≥ 𝒍𝒍, and let 𝑎𝑎𝑘𝑘=𝑓𝑓(𝑘𝑘), for 𝑘𝑘= 1,2,3, … .Then � 𝑎𝑎𝑘𝑘∞𝑘𝑘=𝑙𝑙and �𝑓𝑓(𝑥𝑥)𝑑𝑑𝑥𝑥∞𝑙𝑙either both converge or both diverge. In the case of convergence, the value of the integral is not, in general, equal to the value of the series. Example.Determine whether the following series converges. �1𝑘𝑘ln𝑘𝑘∞𝑘𝑘=2
|M a t h 1 0 2 | S e c 8 . 3| Page3 Harmonic Series: The series �1𝑘𝑘= 1 +12+13+14+15+16+⋯𝑛𝑛𝑘𝑘=1is known as the harmonic series. Example.Find the value(s) of 𝑝𝑝so that �1𝑘𝑘𝑝𝑝∞𝑘𝑘=1Is convergent.
|M a t h 1 0 2 | S e c 8 . 3| Page4 The 𝒑𝒑-series: The following series, which known as the 𝒑𝒑-series, �1𝑘𝑘𝑝𝑝∞𝑘𝑘=1convergesfor 𝑝𝑝> 1and divergesfor 𝑝𝑝 ≤1. Examples:a) �1𝑘𝑘3∞𝑘𝑘=1b) � 𝑘𝑘−2/3∞𝑘𝑘=1c) �10√𝑘𝑘43∞𝑘𝑘=1The Comparison TestLet ∑𝑎𝑎𝑘𝑘and ∑𝑏𝑏𝑘𝑘be infinite series with positive terms, and let 0 <𝑎𝑎𝑘𝑘≤ 𝑏𝑏𝑘𝑘.1.If ∑𝑏𝑏𝑘𝑘converges, then ∑𝑎𝑎𝑘𝑘converges. 2.If ∑𝑎𝑎𝑘𝑘diverges, then ∑𝑏𝑏𝑘𝑘diverges. Example.Determine whether the following series converges or diverges: �|cos𝑘𝑘|𝑘𝑘2∞𝑘𝑘=1
|M a t h 1 0 2 | S e c 8 . 3| Page5 Example.Determine whether the following series converges or diverges: �1𝑘𝑘3+ 100∞𝑘𝑘=2Example.Determine whether the following series converges or diverges: �5𝑘𝑘+ 12𝑘𝑘−1∞𝑘𝑘=2
|M a t h 1 0 2 | S e c 8 . 3| Page6 Fact: 1-ln𝑘𝑘 ≤ 𝑛𝑛𝑘𝑘1𝑛𝑛,for integers 𝑘𝑘,𝑛𝑛 ≥12-1≤ln𝑘𝑘,for each integer 𝑘𝑘 ≥3.Example.Determine whether the following series converges or diverges: �ln𝑘𝑘𝑘𝑘12∞𝑘𝑘=2Example:Determine whether the following series converges or diverges: �ln𝑘𝑘𝑘𝑘52∞𝑘𝑘=2
|M a t h 1 0 2 | S e c 8 . 3| Page7 The Limit Comparison TestLet ∑𝑎𝑎𝑘𝑘and ∑𝑏𝑏𝑘𝑘be infinite series with positive terms, and let 𝐿𝐿= lim𝑘𝑘→∞𝑎𝑎𝑘𝑘𝑏𝑏𝑘𝑘.1.If 0 <𝐿𝐿<∞, then ∑𝑎𝑎𝑘𝑘and ∑𝑏𝑏𝑘𝑘either both converge, or both diverge. 2.If 𝐿𝐿= 0and ∑𝑏𝑏𝑘𝑘converges, then ∑𝑎𝑎𝑘𝑘converges. 3.If 𝐿𝐿=∞and ∑𝑏𝑏𝑘𝑘diverges, then ∑𝑎𝑎𝑘𝑘diverges. Example.Determine whether the following series converges or diverges: �𝑘𝑘2+𝑘𝑘+ 1𝑘𝑘3+ 1∞𝑘𝑘=1
|M a t h 1 0 2 | S e c 8 . 3| Page8 Example.Determine whether the following series converges or diverges: �12𝑘𝑘+ 1∞𝑘𝑘=1
|M a t h 1 0 2 | S e c 8 . 3| Page9 Problems to be done at home:Example.Determine whether the following series converges or diverges: �ln𝑘𝑘𝑘𝑘2∞𝑘𝑘=2Example.Determine whether the following series converges or diverges: ��𝑘𝑘𝑘𝑘3+ 1∞𝑘𝑘=1
|M a t h 1 0 2 | S e c 8 . 3| Page10 Example.Determine whether the following series converge or diverges. �1(𝑘𝑘 −1)2∞𝑘𝑘=4Hint: Integral test (convergent) Example.Determine whether the following series converge. �1𝑘𝑘2+ 4∞𝑘𝑘=0Hint: Integral test (convergent)
|M a t h 1 0 2 | S e c 8 . 3| Page11 Example.Determine whether the following series converge. � 𝑘𝑘−23𝑘𝑘∞𝑘𝑘=4Hint: Divergent test (divergent) Solution: Example.Determine whether the following series converge. �2𝑘𝑘𝑘𝑘2∞𝑘𝑘=2Hint: Divergent test (divergent)
|M a t h 1 0 2 | S e c 8 . 3| Page12 Example.Determine whether the following series converge. �1𝑛𝑛ln2𝑛𝑛∞𝑘𝑘=2Hint: Integral test (convergent)