9.2 Series and Convergence

1. 9.2 Series and Convergence Riverfront Park, Spokane, WA

2. Start with a square one unit by one unit: 1 This is an example of an infinite series. 1 This series converges (approaches a limiting value.) Many series do not converge:

3. The infinite series for a sequence is written as: . . .

4. If Sn has a limit as , then the series converges, otherwise it diverges. In an infinite series: a1, a2,… are terms of the series. an is the nth term. Partial sums: nth partial sum

5. Ex. 1 Find the nth partial sum of the series: What’s the pattern for each sum?? The denoms are all and the numerators??

6. Then what is the infinite sum of the series?? 1 So the series converges to 1. This is called a geometric series.

7. This infinite sum converges to if , and diverges if . is the interval of convergence. Geometric Series: In a geometric series, each term is found by multiplying the preceding term by the same number, r, called the common ratio.

8. a r Example 2:

9. a r Example 3:

10. The nth partial sum of a geometric series is: This is a handy formula to know, as you may be asked to find an nth partial sum of a geometric series! From this, you can see where formula for the infinite sum comes from: 0

11. Ex. 4 Find the partial sum of the series: Write out a few terms: Since the middle terms zero out, we are left with: The series converges to 1.

12. The last problem is an example of a telescoping series: Since the middle terms will always cancel out, we can find the partial sum of a telescoping series by:

13. Ex. 5 Find the sum of the series: This can be written in telescoping form using partial fractions: So the nth partial sum is:

14. Ex. 6 a) Find the sum (if it exists) of the series: Re-write: Since we know that the series converges to the sum of: b) Find the sum (if it exists) of the series: Since the series diverges.

15. Ex. 7 Use a geometric series to express as the ratio of two integers. (as a rational number) Note that we can write as:

16. nth Term Test for Divergence The first requirement for convergence of a series is that the terms of the sequence must approach zero. That is, If converges, then The contrapositive of this fact leads us to a useful test for divergence: If , then diverges. (Notice we are talking about the limit of the sequence, not the partial sum!) So how could we use this test? Let’s take a look…

17. Ex. 8 Determine if the series diverges: a) Let’s look at the sequence: Since the limit of the sequence is not 0, by the nth term test, the series diverges! b) Since the limit of the sequence is not 0, by the nth term test, the series diverges! The limit of the nth term is 0, so… the test fails! We can’t draw a conclusion about convergence or divergence. c) 0

18. Bouncing Ball Problem A ball is dropped from a height of 6 feet and begins bouncing. The height of each bounce is three-fourths the height of its previous bounce. Find the total vertical distance traveled by the ball. Let D1 be the initial distance the ball travels: Let D2 be the distance traveled up and down:

19. Notice this is a geometric series! 