If the sequence is convergent and exists as a real number, then the series is called convergent and we write The number s is called the sum of the series. Otherwise, the series is called divergent. | Module 1: Series Duy Tân University Lecturer: Thân Thị Quỳnh Dao Natural Sciences Department Chapter 3: Series Module 1: Series 1. Definition Let be an infinite sequence. Then, 1. Definition Let be an infinite sequence. Then, Let: 1. Definition If the sequence is convergent and exists as a real number, then the series is called convergent and we write The number s is called the sum of the series. Otherwise, the series is called divergent. Example: Are the following series convergent or divergent? Steps to determine the convergence or divergence of series: - Caculate sn - Find lim sn + lim sn = s + lim sn = + Don’t exist the limit of sn - Determine an Special series: : geometric series. : p- series. test: The Test for Divergent The Comparison test. The Limit Comparison test. The Alternating Series test. The Ration test. The Root test. test: a. The Test for Divergence. Let . If or does not exist then the series is divergent tests: b. The Comparison test. Positive serie: is called positive serie if tests: The Comparison test: Let are positive series. If then, either both convergent or both divergent. b. The Comparison test. tests: b. The Comparison test. Some special series: : geometric series : p- series Convergent if Divergent if Convergent if Divergent if test: c. The Ratio test. Let : and If then divergent. If then convergent. | Module 1: Series Duy Tân University Lecturer: Thân Thị Quỳnh Dao Natural Sciences Department Chapter 3: Series Module 1: Series 1. Definition Let be an infinite sequence. Then, 1. Definition Let be an infinite sequence. Then, Let: 1. Definition If the sequence is convergent and exists as a real number, then the series is called convergent and we write The number s is called the sum of the series. Otherwise, the series is called divergent. Example: Are the following series convergent or divergent? Steps to determine the convergence or divergence of series: - Caculate sn - Find lim sn + lim sn = s + lim sn = + Don’t exist the limit of sn - Determine an Special series: : geometric series. : p- series. test: The Test for Divergent The Comparison test. The Limit Comparison test. The Alternating Series test. The Ration test. The Root test. test: a. The Test for Divergence. Let . If or does not exist then the series is divergent tests: b. The Comparison test. Positive serie: is called positive serie if tests: The Comparison test: Let are positive series. If then, either both convergent or both divergent. b. The Comparison test. tests: b. The Comparison test. Some special series: : geometric series : p- series Convergent if Divergent if Convergent if Divergent if test: c. The Ratio test. Let : and If then divergent. If then convergent.
