A list of numbers following a certain pattern

{an} = a1 , a2 , a3 , a4 , … , an , …

Pattern is determined by position or by what has come before

3, 6, 12, 24, 48, …

,...5,3,2,1,1

Lecture 21 – Sequences

1

nan 25

Find the first four terms and the 100th term for the following:

3

)1()1(

n

na

n

n

Defined by n(position)

2

An arithmetic sequence is the following:

... ,4,3,2,, dadadadaa

:is sequence arithmetic then the,7 and 5 If da

with a as the first term and d as the common difference.

Arithmetic Sequence

3

A geometric sequence is the following:

... ,,,,, 432 arararara

Geometric Sequence

4

:is sequence geometric then the,5 and 4 If ra

with a as the first term and r as the common ratio.

. also is }{ sequence theoflimit

then the,)(lim and )( If

La

Lxfnfa

n

xn

Convergence

We say the sequence “converges to L” or, if the sequence

does not converge, we say the sequence “diverges”.

A sequence that is monotonic and bounded converges.

5

Monotonic and Bounded

Monotonic: sequence is non-decreasing (non-increasing)

Bounded: there is a lower bound m and upper bound M such that

6

n allfor Mam n

Monotonic & Bounded:

Monotonic & not Bounded:

Not Monotonic & Bounded:

Not Monotonic & not Bounded:

13

1

n

n an

Example 1 – Converge/Diverge?

7

n

n an

1

Example 2 – Converge/Diverge?

nn e

n a

Example 3 – Converge/Diverge?

8

}{}!{}{}{}){(ln nnpq nnbnn

Growth Rates of Sequences: q, p > 0 and b > 1

n

n

n b

alim

Lecture 22 – Sequences & Series

n

n n a

21

Example 4 – Converge/Diverge?

9

n

n n

21lim

Partial Sums

Adding the first n terms of a sequence, the nth partial sum:

n

kka

1n4321n a ... aaaaS

10

Series – Infinite Sums

If the sequence of partial sums converges, then the series

1

n4321 a aaaak

ka

converges.

Find the first 4 partial sums and then the

nth partial sum for the sequence defined by: n4

3a n

11 aS

212 aaS

3213 aaaS

aaaaS 43214

Sn 11

Example 1

The partial sum for a geometric sequence looks like:1-n32

n ar ... arararaS

sum infinite theand , as 0 then 1, || If nrr n

...ar ... ararara 1-n32

1432 nararararara

12

Geometric Series

Find the sum of the geometric series:

...125

8

25

4

5

2

...49

1

7

11

r

aS

1

...12121212.

Geometric Series – Examples

13

Lecture 23 – More Series

Find the sum of the geometric series:

k

k 0

1

r

aS

1Geometric Series – More Examples

14

1k

ke

21

1

4

3

kk

k

15

Telescoping Series – Example 1

0 )43)(13(

1

k kk

130

1

70

1

28

1

4

1

16

Telescoping Series – Example 1 – continued

n

kn kk

S0 )43)(13(

1

Telescoping Series – Example 2

17

1 2ln

k k

k

6

4ln

5

3ln

4

2ln

3

1ln

...9

1

4

11

1

12

k k...

125

3

25

3

5

3

5

3

1

kk

...4

1

3

1

2

11

1

1

k k

49138888.1

46361111.1

42361111.1

36111111.1

25.1

1

6

5

4

3

2

1

S

S

S

S

S

S

749952.

74976.

7488.

744.

72.

6.

6

5

4

3

2

1

S

S

S

S

S

S

45.2

283333.2

083333.2

833333.1

5.1

1

6

5

4

3

2

1

S

S

S

S

S

S

______lim n

nS ______lim

nn

S ______lim n

nS

18