9.1 Part 19.1 Part 1Sequences and SeriesSequences and Series

?na

Write a general expression for the following sequential terms:Write a general expression for the following sequential terms:

This is called an This is called an arithmetic sequencearithmetic sequence

As As nn goes to infinity, what does goes to infinity, what does aann approach? approach?

2

11 a

4

12 a

8

13 a ?4 a ?5 a

0

1

1

Does the sum of the terms:Does the sum of the terms:

approach a limiting value?approach a limiting value?

Use the 1 X 1 square above Use the 1 X 1 square above to find your answer.to find your answer.

1

2

1

4

1

8

1

16

1

32

1

64 ...

1 2

1

n

n

1

4

This is an example of an This is an example of an infinite seriesinfinite series..

1

1

You can start with this You can start with this one by one square:one by one square:

1

21

21

4

1

81

8

1

161

16

1

32 1

64

1

32

1

64 1

This series This series convergesconverges (approaches a limiting value.) (approaches a limiting value.)

Since the area of all of these Since the area of all of these slices of the square add up slices of the square add up to the area of the square…to the area of the square…

A ball bounces with elasticity such that each time it bouncesA ball bounces with elasticity such that each time it bouncesit returns to half it’s previous height. If the ball was dropped it returns to half it’s previous height. If the ball was dropped from a height of 5 feet, find the total distance that the ball from a height of 5 feet, find the total distance that the ball travels.travels.

A ball bounces with elasticity such that each time it bouncesA ball bounces with elasticity such that each time it bouncesit returns to half it’s previous height. If the ball was dropped it returns to half it’s previous height. If the ball was dropped from a height of 5 feet, find the total distance that the ball from a height of 5 feet, find the total distance that the ball travels.travels.

5 52

1

5

2

1

5

2

1

2

1

5

2

1

2

1

…

1 2

110

n

n

52

12

5

2

1

2

12

5

We’ll come back to this…is there a pattern that you We’ll come back to this…is there a pattern that you are seeing here?are seeing here?

What seems to be What seems to be the answer you are the answer you are all getting?all getting?

In an infinite series:In an infinite series: 1 2 31

n kk

a a a a a

aa11, a, a22,…,… are are termsterms of the series. of the series. aann is the is the nnthth term term..

Partial sums:Partial sums: 1 1S a

2 1 2S a a

3 1 2 3S a a a

1

n

n kk

S a

nnthth partial sum partial sum

If If SSnn has a limit as , then the series converges, has a limit as , then the series converges,

otherwise it otherwise it divergesdiverges..

n

In a In a geometric seriesgeometric series, each term is found by multiplying , each term is found by multiplying

the preceding term by the same number, the preceding term by the same number, rr. Now we want . Now we want to see if we can find a short formula for to see if we can find a short formula for

Geometric Series:

2 3 1 1

1

n n

n

a ar ar ar ar ar

nSnS

nn arararararrS 432

Subtract the second line from the first. What do you get?Subtract the second line from the first. What do you get?

Sn rSn a arn

Can you determine the limit from this formula? Can you determine the limit from this formula?

The partial sum of a geometric series is:The partial sum of a geometric series is: 1

1

n

n

a rS

r

If thenIf then1r 1

lim1

n

n

a r

r

1

a

r

0

Geometric Series:

In a In a geometric seriesgeometric series, each term is found by multiplying , each term is found by multiplying

the preceding term by the same number, the preceding term by the same number, rr..

2 3 1 1

1

n n

n

a ar ar ar ar ar

This converges to if , and diverges if .This converges to if , and diverges if .1

a

r1r 1r

1 1r is the is the interval of convergenceinterval of convergence..

Geometric Series:

In a In a geometric seriesgeometric series, each term is found by multiplying , each term is found by multiplying

the preceding term by the same number, the preceding term by the same number, rr..

2 3 1 1

1

n n

n

a ar ar ar ar ar

This converges to if , and diverges if .This converges to if , and diverges if .1

a

r1r 1r

0n

narororA geometric A geometric series can be series can be expressed asexpressed as

As long as the As long as the first exponent first exponent is of is of rr is is 00

Example 1:Example 1:

3 3 3 3

10 100 1000 10000

.3 .03 .003 .0003 .333...

1

3

310

11

10

aa

rr

3109

10

3

9

1

3

1

1

10

1

10

3

n

n

1 1 11

2 4 8

11

12

11

12

13

2

2

3

aa

rr

Example 2:Example 2:

1

1

2

1

n

n

Now let’s return to the bouncing ball that bouncesNow let’s return to the bouncing ball that bouncesto half it’s previous height each time it bounces. If the ball to half it’s previous height each time it bounces. If the ball was dropped from a height of 5 feet, find the total distance was dropped from a height of 5 feet, find the total distance that the ball travels.that the ball travels.

1

1

2

155

n

n

21

1

55 feet15

1 2

110

n

n

5

1

1

2

1

2

1105

n

n