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?

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

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?

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

Example #1

You Try! Find the sum of this series:

Answer  

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 total distance it travels before coming to rest?

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”.

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:

Possible Answers to Infinite Limits You may get zero or any number.

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.

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?