1. Natural Sciences Department
Duy Tân University
Natural Sciences Department
Module 1:
Series
Lecturer: Thân Thị Quỳnh Dao
Chapter 3: Series Module 1: Series

2. 1. Definition
Let be an infinite sequence.
Then,

3. 1. Definition
Let be an infinite sequence.
Then,
Let:

4. 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.

5. Example:
Are the following series convergent or divergent?

6. Steps to determine the convergence or divergence of series:
- Determine an
- Caculate sn
- Find lim sn
+ lim sn = s
+ lim sn =
+ Don’t exist the limit of sn

7. Special series:
: geometric series.
: p- series.

8. 2.Some test:
The Test for Divergent
The Comparison test.
The Limit Comparison test.
The Alternating Series test.
The Ration test.
The Root test.

9. 2.Some test:
a. The Test for Divergence.
Let
If or does not exist then the series
is divergent

10. 2.Some tests:
b. The Comparison test.
Positive serie:
is called positive serie if

11. 2.Some tests:
b. The Comparison test.
The Comparison test: Let are positive series. If
then, either both convergent or both divergent.

12. 2.Some tests:
b. The Comparison test.
Some special series:
Convergent if
: geometric series
Divergent if
Convergent if
: p- series
Divergent if

13. 2.Some test:
c. The Ratio test.
Let : and
If then divergent.
If then convergent.