1. Series and Convergence
Lesson 9.2

2. Definition of Series
Consider summing the terms of an infinite sequence
We often look at a partial sum of n terms

3. Definition of Series
We can also look at a sequence of partial sums { Sn }
The series can converge with sum S
The sequence of partial sums converges
If the sequence { Sn } does not converge, the series diverges and has no sum

4. Examples
Convergent
Divergent

5. Telescoping Series Consider the series
Note how these could be regrouped and the end result
As n gets large, the series = 1

6. Geometric Series Definition Example An infinite series
The ratio of successive terms in the series is a constant
Example
What is r ?

7. Properties of Infinite Series
Linearity
The series of a sum = the sum of the series
Use the property

8. Geometric Series Theorem
Given geometric series (with a ≠ 0)
Series will
Diverge when | r | ≥ 1
Converge when | r | < 1
Examples
Compound interest Or

9. Geometric Series Theorem
It can be shown that
Try on our example:
a = ?
r = ?

10. Applications – Pendulum Swing

11. Applications – Pendulum Swing
We will assume
Original distance of 3.55
Decay of .87 each time
What is the series?
What is the sum of the series?

12. Assignment
Lesson 9.2
Page 612
Exercises 1 – 69 EOO