1. 9.2 Series & Convergence Objectives:
Understand convergent infinite series
Use properties of infinite geometric series
Use the nth term test for divergence
©2004Roy L. Gover (

2. Definition
An infinite series is a sum that continues without end. It is represented by:
sometimes n=0
where are terms of the series.

3. Definition
A sequence is an ordered listing of numbers separated by commas. It is represented by:
where are terms of the sequence.

4. Definition
The sequence:
where
is a sequence of partial sums.

5. Definition
A sequence or series converges if its terms approach a value. If the terms do not approach a value, the sequence diverges.

6. Important Idea
If the sequence of partial sums converge to S, then the infinite series converges and its sum is S. If the sequence of partial sums diverge, the series diverges and has no sum.

7. Example 1 Consider a 1 x 1 square…
A convergent series has a limiting value
Series that do not converge are said to diverge
An example of an infinite series that converges to 1 1

8. Example
The series:
has the following partial sums:

9. Example (cont.) The sequence of partial sums:
Do not confuse the sequence of partial sums with the original sequence which is summed to make a series

10. Example (cont.)
Find the nth term (pattern) of the sequence of partial sums:
and

11. Example (cont.)
Since:
the original series converges and has a sum of 1.

12. Example Find the sum of the series, if it exists:
Hint: find the limit of the nth partial sum.

13. Example (cont) The sum is 1
Note:This is called a telescoping series. Why?

14. Try This
Find the sum, if it exists:
No limit-the sequence diverges

15. Definition
is a geometric series with common ratio r

16. Important Idea Theorem: A geometric series converges (sums) to:
if Otherwise, the series diverges and has no sum.

17. Try This r a Find the sum (if it exists) of the geometric series: 6
Can you confirm with your graphing calculator?

18. Try This Does the following series converge or diverge? converge
Convergence is not affected by removal of a finite number of terms from the beginning of the series.
a/(1-r)-(3+3/2+3/4)

19. c is any real number, then:
Important Idea
Theorem:
If and and
c is any real number, then:
&

20. Important Idea
Theorem:
If a series converges, the limit of its nth term is 0.
Note: This theorem does not state that if the limit of the nth term is 0, the series converges.

21. Example nth term The series: converges therefore:
This idea is not very helpful, but its contrapositive is …

22. Important Idea
Theorem- nth Term Test for Divergence:
If the limit of the nth term is not 0, then the series diverges
Problem:sometimes it is difficult to find the nth term

23. Example
diverges because

24. Example For the series:
, and you can draw no conclusion about convergence or divergence.