Day 4 Notes Infinite Geometric Series

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 decrease each time, the sum gets closer and closer to a specific value.

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

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

If a sum exists, how is it found? The first term is a 1 and the common ratio is r, to find the sum of an infinite geometric series use the following formula:

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

Finding a common ratio Given the sum, S, and the first term, plug in what you know and solve for r. a) S = ; a 1 = 5b) S = -2; a 1 = - Cross multiply 25 = 27 – 27rSolve for r r = Cross multiply -4/3 = -2+ 2rSolve for r r = WATCH and COPY down TRY on your own and CHECK it.

Finding the First Term Given the sum, S, and the common ratio, r, plug in what you know and solve for a 1. a) S = 54; r = 0.2b) S = 2; r = Simplify denominator a 1 = 43.2 WATCH and COPY down Get a 1 by itself by multiplying both sides by denominator. Simplify denominator a 1 = TRY on your own and CHECK it. Get a 1 by itself by multiplying both sides by denominator.

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)

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

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