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Warm-up:

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

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Warm-up:

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  1. Warm-up: • p 185 #1 – 7

  2. 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?

  3. 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?

  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?

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

  6. Example #1 • Find the sum of the series:

  7. 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?

  8. Example #3 • Write 0.123123123… as a fraction using an Infinite Geometric Series.

  9. Try another… • Write 12.77777777…as a fraction using a geometric series.

  10. Limits • Limits are used to determine how a function, sequence or series will behave as the independent variable approaches a certain value, often infinity.

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

  12. 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:

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

  14. 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.

  15. 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.

  16. 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. Let’s do some test values…

  17. 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. Let’s do some test values… This approaches 1/3 but how do I prove it?

  18. 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

  19. 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).

  20. Limits • Use the fact that to evaluate the following:

  21. 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?

  22. Homework: • P 781 # 15 – 39 odd, 40

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