section 8.2: infinite series. zeno’s paradox can you add infinitely many numbers ?? you can’t...

Section 8.2: Infinite Series

Zeno’s Paradox

Can you add infinitely many numbers ??

1

2

1

4

1...

8 ? ?

You can’t actually get anywhere because you always have to cover half

the remaining distance!

You have to do half, then half that.. etc.

Informal Definition

A series is sequence added up.

A series converges

if the sequence of partial sums converges.

Formal Definition

1k

k

a

1

n

n kk

S a

Example

1

1

2

k

k

1

n

n kk

S a

1

2

1

4

1...

8

1

1

2S

4

1 1 1 1

2 4 8 16S

3

1 1 1

2 4 8S

2

1 1

2 4S

2

0

...k

k

ar a ar ar

Definition

is a geometric series.

0

k

k

ar

Theorem

Diverges if

Converges to if

| 1|r

1

a

r| 1|r

Proof:

nS

nr S

(1 ) nr S

nS

0

k

k

ar

... na ar ar 1... n nar ar ar

a

1

(1 )

na ar

r

1

a

r

limn

limn

If and only if| | 1r

1nar

0

2

3

k

k

12

13

3

0

1

2

k

k

1 1 1

2 41

8 2

11

12

1

1

2

k

k

1 1 1

2 4 8 1

112

1

This solves Zeno’s paradox!!

1

2

283

k

k

1

2

243

k

kk

0

283

k

k

2

2 243

* k

kk

2

283k

k

81

82

13

24

2

2824

383

28

k

k

32

3

283

1

12

415

k k

kk

2

420

5

k

k

20204

15

100

9

2

100

9

420 2

420

550

k

k

2

54 * 4

15

k k

kk

0

420

5

k

k

420

5

2

420

5

k

k

64

9

11

372

k

kk

1

314

2

k

k

Diverges

Telescoping Series

1

1

( 1)k k k

Use Partial Fractions1

1 1

( 1)k k k

nS 1 1

2 3

112

1 1...

1n n

11

1n

1 1

3 4

lim nnS

1

1

( 1)k k k

11lim 1

1n n