goal: does a series converge or diverge? lecture 24 – divergence test 1 divergence test (if a...

Goal: Does a series converge or diverge?

Lecture 24 – Divergence Test

1

?1

kka

Divergence Test (If a series converges, then sequence converges to 0.)

1 13

12

k k

kExample 1 – Converge/Diverge?

2

Example 2 – Converge/Diverge?

1

1tank

k

0

1

3

2

kk

k

Example 3 – Converge/Diverge?

3

0

1

3

2

kk

k

However,

1

1

n n

Example 4 – Converge/Diverge?

4

1

1

n n

9

1

8

1

7

1

6

1

5

1

4

1

3

1

2

11

However,

Goal: Does a series converge or diverge?

then, allfor function decreasing

positive, ,continuous a with )( If

Nx

fnfan

Lecture 25 – Integral Test

5

?1

12

n n

Integral Test (The area under a function and infinite sum of the terms in a sequence defined by that function are related.)

N a

6

If area under curve is bounded,

then so is

But then is a bounded,

monotonic sequence.

So it converges and

thus converges.

1N a2N a

1N N 1N

N a

If area under curve is unbounded,

then

is also unbounded.

And thus, diverges.

1N a2N a

1N N 1N

12

1

n n

Example 1 – Converge/Diverge?

7

2

1lim

n

n

14

19

1

1 2 3

0 sequence converges to zero.

No info from Divergence Test.

1

1

nne

Example 2 – Converge/Diverge?

8

nn e

1lim 0

n

n e1

1

sequence converges to zero.

No info from Divergence Test.

2

ln

n n

nExample 3 – Converge/Diverge?

9

n

n

n

lnlim 0 sequence converges to zero.

No info from Divergence Test.

2

2ln

3

3ln

4

4ln

1 2 3

021

1

k k

Example 4 – Converge/Diverge?

10

21

1lim

n

n0 sequence converges to zero.

No info from Divergence Test.

11

Lecture 26 – Ratio and Root Tests

Goal: Does a series of positive terms converge or diverge? ?1

kka

Ratio Test (Does ratio of successive terms approach some limit L? Then series is close to being geometric.)

,

,

,

1

1

1

lim 1

L

L

L

a

a

n

n

n

12

Root Test (Does nth root of terms approach some limit L? Then series is close to being geometric.)

,

,

,

1

1

1

lim

L

L

L

ann

n

2 ln

1

nnn

Example 1 – Converge/Diverge?

13

nn n

ln

1lim 0 sequence converges to zero.

No info from Divergence Test.

1

2

1n

n

n

k

Example 2

14

For what values k does the series converge?

n

n

n n

k2

1lim

1

nn

n n

k/12

1lim

n

n n

k

1lim

n

kn

n1lnlim

2

3

3nn

n

Example 3 – Converge/Diverge?

15

nn

n

3lim

3

0 sequence converges to zero.

No info from Divergence Test.

0

2

! 2

!

k k

k

Example 4 – Converge/Diverge?

16

1.

2.

Direct Comparison:

17

Lecture 27 – Comparison Tests

N a

1N a2N a

1N N

N a

1N a2N a1N N 1N

converges. then converges, and If

Nn

nNn

nnn abba

diverges. then diverges, and If

Nn

nNn

nnn abba

N b1N b

2N b

N b

1N b 2N b

Limit Comparison:

18

terms.positive with series be and Let

Nnn

Nnn ba

diverge.both or convergeboth

and then 0,withlim Ifn

Nn

nNn

nn

n ba LL b

a

1 15

5

n n

Example 1 – Converge/Diverge?

19

15

5lim

n

n0 sequence converges to zero.

No info from Divergence Test.

, as n

Example 2 – Converge/Diverge?

20

1 !

1

n n

!

1lim

n

n0 sequence converges to zero.

No info from Divergence Test.

:2for general,In n

:3n :4n

Example 3 – Converge/Diverge?

21

22 1

1

n n

1

1lim

2n

n0 sequence converges to zero.

No info from Divergence Test.

22

1

1

1, as

nnn

Example 4 – Converge/Diverge?

22

12 1

1

n n

1

1lim

2n

n0 sequence converges to zero.

No info from Divergence Test.

nnnn

11

1

1, as

22