Warm-up:

p 185 #1 – 7

Section 12-3: Infinite Sequences and Series

In this section we will answer…What makes a sequence infinite?How can something infinite have a

limit? Is it possible to find the sum of an

infinite series?

What kind of sequence is it?Find the 18th term.Now find the 20th, 25th, and 50th. So …the larger n is the more the

sequence approaches what?

Consider the following sequence:16, 8, 4, ….

Sum of an Infinite Geometric Series

In certain sequences, as n increases, the terms of the sequence will decrease, and ultimately approach zero.

This occurs when ______________.

What will happen to the Sum of the Series?

1

1n

aS

r

1

1n

aS

r

Sum of an Infinite Geometric Series

The sum, Sn, of an infinite geometric series for which is given by the following formula:

1

1n

aS

r

1

1n

aS

r

1

1n

aS

r

1r

Example #1

Find the sum of the series:

1 1 1...

25 250 2500

Example #2

A tennis ball dropped from a height of 24 feet bounces 50% of the height from which it fell on each bounce. What is the vertical distance it travels before coming to rest?

Example #3

Write 0.123123123… as a fraction using an Infinite Geometric Series.

Try another…

Write 12.77777777…as a fraction using a geometric series.

Limits Limits are used to determine how a

function, sequence or series will behave as the independent variable approaches a certain value, often infinity.

Limits They are written in the form below: It is read “The limit of 1 over n as n approaches

infinity”.

1limn n

Limits They are written in the form below: It is read “The limit of 1 over n as n approaches

infinity”. To evaluate the limit substitute infinity for n:

1 1lim 0n n

Possible Answers to Infinite Limits

You may get zero or any number.

1

2lim

5

2lim 6

nn

n n

Possible Answers to Infinite Limits

You may get infinity. That means no limit exists because it does not approach

any single value.

You may get no limit exists because the sequence fluctuates.

lim3n

n

( 1)lim

n

x n

Possible Answers to Infinite Limits

You may get infinity over infinity. This is indeterminate; meaning in its present

form you can’t tell if it has a limit or not.

3

3

6lim

3n

n nn

Possible Answers to Infinite Limits

You may get infinity over infinity. This is indeterminate; meaning in its present

form you can’t tell if it has a limit or not.

3

3

6lim

3n

n nn

Let’s do some test values…

Possible Answers to Infinite Limits

You may get infinity over infinity. This is indeterminate meaning in its present

form you can’t tell if it has a limit or not.

3

3

6lim

3n

n nn

Let’s do some test values…

This approaches 1/3 but how do I prove it?

Algebraic Manipulation of Limits

Method 1: Works only if denominator is a single term.– 1) If denominator is single term, split the into

separate fractions.– 2) Reduce– 3) Take Limit

3

3

6lim

3n

n nn

Algebraic Manipulation of Limits

Method 2: This works for all infinite limits.– 1) Divide each part of the fraction by the highest

power of n shown.– 2) Reduce.– 3) Take limit (Some terms will drop out).

4 3

4

3 1lim

2 7n

n nn n

Limits

Use the fact that to evaluate the following:

3

3

6lim

3n

n n

n

1lim 0n n

3

3

6lim

3n

n n

n

3

4 2limn

n

n

23 4lim

2

n

n

n

2

2

3 4lim

1n

n n

n

2

2

3 2 5lim

4 1

n

n n

n

The Recap:

What makes a sequence infinite?

How can something infinite have a limit?

Is it possible to find the sum of an infinite series?

Homework:

P 781 # 15 – 39 odd, 40