9.2 Series & Convergence Objectives: Understand convergent infinite series Use properties of infinite geometric series Use the nth term test for divergence ©2004Roy L. Gover (www.mrgover.com)

Definition An infinite series is a sum that continues without end. It is represented by: sometimes n=0 where are terms of the series.

Definition A sequence is an ordered listing of numbers separated by commas. It is represented by: where are terms of the sequence.

Definition The sequence: where is a sequence of partial sums.

Definition A sequence or series converges if its terms approach a value. If the terms do not approach a value, the sequence diverges.

Important Idea If the sequence of partial sums converge to S, then the infinite series converges and its sum is S. If the sequence of partial sums diverge, the series diverges and has no sum.

Example 1 Consider a 1 x 1 square… A convergent series has a limiting value Series that do not converge are said to diverge An example of an infinite series that converges to 1 1

Example The series: has the following partial sums:

Example (cont.) The sequence of partial sums: Do not confuse the sequence of partial sums with the original sequence which is summed to make a series

Example (cont.) Find the nth term (pattern) of the sequence of partial sums: and

Example (cont.) Since: the original series converges and has a sum of 1.

Example Find the sum of the series, if it exists: Hint: find the limit of the nth partial sum.

Example (cont) The sum is 1 Note:This is called a telescoping series. Why?

Try This Find the sum, if it exists: No limit-the sequence diverges

Definition is a geometric series with common ratio r

Important Idea Theorem: A geometric series converges (sums) to: if . Otherwise, the series diverges and has no sum.

Try This r a Find the sum (if it exists) of the geometric series: 6 Can you confirm with your graphing calculator?

Try This Does the following series converge or diverge? converge Convergence is not affected by removal of a finite number of terms from the beginning of the series. a/(1-r)-(3+3/2+3/4)

c is any real number, then: Important Idea Theorem: If and and c is any real number, then: &

Important Idea Theorem: If a series converges, the limit of its nth term is 0. Note: This theorem does not state that if the limit of the nth term is 0, the series converges.

Example nth term The series: converges therefore: This idea is not very helpful, but its contrapositive is …

Important Idea Theorem- nth Term Test for Divergence: If the limit of the nth term is not 0, then the series diverges Problem:sometimes it is difficult to find the nth term

Example diverges because

Example For the series: , and you can draw no conclusion about convergence or divergence.