1 / 11

Day 4 Notes

Learn how to find the sum and common ratio of an infinite geometric series. Understand when a sum exists and how to write a repeating decimal as a fraction.

erwinm
Download Presentation

Day 4 Notes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Day 4 Notes Infinite Geometric Series

  2. The Sum of an Infinite Geometric Series If the list of terms goes on infinitely, how is it possible to add them together if it never stops? Mathematically, if a geometric sequence has a pattern where terms decreaseeach time, the sum gets closer and closer to a specific value.

  3. When does a sum exist? • when | r | < 1 • when the terms decrease each time (ignoring negative terms)

  4. Some vocab… A series converges if it does have a sum. If | r |  1, the series has no sum. A series diverges if it does not have a sum. We answer “the sum diverges” when | r |  1

  5. If a sum exists, how is it found? The first term is a1 and the common ratio is r, to find the sum of an infinite geometric series use the following formula:

  6. Example 1: Find the sum of the infinite geometric series, if possible. a) b) c) d) e) TRY on your own and CHECK it. WATCH and COPY down r = .1, so r < 1, so a sum exists. r = 1/3, so r < 1, so a sum exists. S = S = S = S = 18 TRY on your own and CHECK it. TRY on your own and CHECK it. r = .6, so r < 1, so a sum exists. r = 5/4, so r > 1, so there is no sum. The sum diverges S = S = 10 r = -1/2, so |r| < 1, so a sum exists. S = S = -20 TRY on your own and CHECK it.

  7. Finding a common ratio Given the sum, S, and the first term, plug in what you know and solve for r. a) S = ; a1 = 5 b) S = -2; a1 = - TRY on your own and CHECK it. WATCH and COPY down Cross multiply Cross multiply -4/3 = -2+ 2r Solve for r 25 = 27 – 27r Solve for r r = r =

  8. Finding the First Term Given the sum, S, and the common ratio, r, plug in what you know and solve for a1. a) S = 54; r = 0.2 b) S = 2; r = TRY on your own and CHECK it. WATCH and COPY down Simplify denominator Simplify denominator Get a1 by itself by multiplying both sides by denominator. Get a1 by itself by multiplying both sides by denominator. a1 = 43.2 a1 =

  9. Writing a Repeating Decimal as a Fraction 1) Find the repeating portion. (it could be 1 digit, 2 digits, 3 digits, etc…) 2) Create a fraction with a denominator of 9’s. (Use as many 9’s as there are repeating digits) 3) Simplify. 4) If there are terms on the left side of the decimal you must use MIXED fractions. ***If there are terms that DON’T repeat, you need to move those numbers to the left of the decimal place, then use mixed fractions. Remember to move the decimal of your denominator back the amount you moved the numbers over. (See example d)

  10. Example 4: Write the repeating decimal as a fraction. TRY on your own and CHECK it. WATCH and COPY down a) 0.6666….. b) 0.327327… 1) There is one repeating digit. 6 repeats. 2) Place the repeating digit over as many 9s needed. 3) Simplify. 1) There are three repeating digits. 327 repeats. 2) Place the repeating digits over as many 9s needed. 3) Simplify.

  11. Example 4, continued… WATCH and COPY down WATCH and COPY down c) 27.2727… d) 0.416666… 4) There are terms on the left side of the decimal. Use a mixed fraction in step 2. 1) There are two repeating digits. 27 repeats. 2) Place the repeating digits over as many 9s needed. (turn to an improper fraction) 3) Simplify. **There are digits that don’t repeat. Move the decimal 2 places to the right. (multiply by 100) 41.6666… 4) There are terms on the left side of the decimal. Use a mixed fraction in step 2. 1) There is one repeating digit. 6 repeats. 2) Place the repeating digits over as many 9s needed. **We must undo the decimal move, so divide by 100. Then reduce. 3) Simplify.

More Related