1. SERIES AND CONVERGENCE
Chapter 9.2
SERIES AND CONVERGENCE

2. After you finish your HOMEWORK you will be able to…
Understand the definition of a convergent infinite series
Use properties of infinite geometric series
Use the nth-Term Test for Divergence of an infinite series

3. INFINITE SERIES
An infinite series (aka series) is the sum of the terms of an infinite sequence.
Each of the numbers, , are called terms of the series.

4. CONVERGENT AND DIVERGENT SERIES
For the infinite series , the n-th partial sum is given by
If the sequence of partial sums, , converges
to , then the series converges. The limit is called the sum of the series.
If diverges, then the series diverges.
Series may also start with n = 0.

5. THE BATHTUB ANALOGY

6. DIVERGE VERSUS CONVERGE
Consider the series
What happens if you continue adding 1 cup of water?
Consider the series
How is this situation different?
Will the tub fill?

7. TELESCOPING SERIES What do you notice about the following series?
What is the nth partial sum?

8. CONVERGENCE OF A TELESCOPING SERIES
A telescoping series will converge if and only if approaches a finite number as n approaches infinity.
If it does converge, its sum is

9. GEOMETRIC SERIES
The following series is a geometric series with ratio r.

10. THEOREM 9.6 CONVERGENCE OF A GEOMETRIC SERIES
A geometric series with ratio r diverges if
If then the series converges to

11. THEOREM 9.7 PROPERTIES OF INFINITE SERIES
If is a real number,
then the following series converge to the
indicated sums.

12. THEOREM 9.8 LIMIT OF nth TERM OF A CONVERGENT SERIES
If converges, then
Why?

13. The nth-Term Test THEOREM 9.9
If , the
infinite series
diverges.