Warm up 1. Find the tenth term in the sequence:

2. Find the sum of the first 6 terms of the geometric series 2-8+32-128…

If r=-2 and a8 = -384 what is the first term of the sequence?

,...33,3,3

12.3 Infinite Sequences and Series

Objective: To find the limit of the terms of an infinite sequence

To find the sum of an infinite geometric series

Infinite Sequences & Series

Infinite sequence – a sequence that has infinitely many terms.

as the “n” increases the terms decrease and get closer to 0. (no term actually becomes 0) But 0 is called the limit of the terms of the sequence.

Limits can be used to determine if a sequence approaches a value.

When any positive power of n appears only in the denominator of a fraction and n approaches infinity, the limit equals zero.

1lim 0

"the limit of 1 over n, as n approaches infinity, equals zero"

n n

Example Estimate the limit of the sequence:

If n=50 3.398447 If n=100 3.448374 If n=500 3.489535 If n=1000 3.494759

nn

n

32

27,...,

27

65,

4

16,

5

92

2

Ex 1Estimate the limit of

1

12...,,

28

17,

9

7,

2

13

2

n

n

01

123

2

lim

n

n

n

Theorems of Limits For sequences with more complicated general forms.

Applications of the following limit theorems can make the limit easier to find.

If the exists, exists, and c is a constant, then the following theorems are true.

Limit of a Sum

Limit of a Difference

Limit of a Product

Limit of a Quotient

Limit of a Constant

nna

lim n

nb

lim

nn

nn

nnn

baba

limlimlim

nn

nn

nnn

baba

limlimlim

nn

nn

nnn

baba

limlimlim

0lim,lim

limlim

nn

nn

nn

n

n

nbwhere

b

a

b

a

neachforccwherecc nnn

,lim

ExampleFind the limit

3 6limx

x

x

Examplefind the limit

2

2

3 4lim

1x

x x

x

Limits don’t exist for all infinite sequences. If the absolute value of a sequence becomes arbitrarily great or if the terms don’t approach a value the sequence has no limit. Example

24 6lim

3x

x

x

Ex 5( 1)

lim5 1

x

x

x

x

When n is even, (-1)n = 1 and when n is odd, (-1)n = -1. Therefore the sequence would have no limit.

Finding the limit If the largest exponents are the same in

the numerator and the denominator, then the limit is the ratio of the coefficients of the terms containing the largest exponent.

If the largest exponent is in the numerator, then there is no limit.

If the largest exponent is in the denominator, then the limit is 0.

Sum of an Infinite Series If Sn is the sum of the first n terms of a

series, and S is a number such that S-Sn approaches zero as n increases without bound, then the sum of the infinite series is S.

lim nnS S

Sum of the Terms of an Infinite Geometric Sequence

The sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where |r| < 1 is given by

.

1

1

aS

r

Sum of an Infinite Geometric Sequence

Examplefind the sum of 60 + 24 + 9.6 + …

1

1

aS

r

Example Write as a fraction

So and

_____

762.0

...000,000,000,1

762

000,000,1

762

1000

762

1000

7621 a

1000

1r

10001

1

1000762

S

333

254

999

762or

Practice

Write as a fraction. _____

981.12