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12-3 – Infinite Sequences and Series

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12-3 – Infinite Sequences and Series Notes 12-3 and 12-5: Infinite Sequences and Series, Summation Notation I. Infinite Sequences and Series A. Concept and Formula 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? • W=When 푟 < 1, as n increases, the terms of the sequence will decrease, and ultimately approach zero. Zero is the limit of the terms in this sequence. • What will happen to the Sum of the Series? It will reach a limit as well. The sum, Sn, of an infinite geometric series for a1 Sn which 푟 < 1 is given by 1 r the following formula: Notice that 푟 < 1. If 푟 > 1, the sum does not exist. The series must also be geometric. Why? 3 Ex 1: Find the sum of the series 21 − 3 + − ⋯ 7 푎 −3 −1 푟 = 2 = = 푎1 21 7 푎 푆 = 1 푛 1 − 푟 21 푆 = 푛 1 1 − (− ) 7 푆푛 = 18.375 Ex 2: Find the sum of the series 60 + 24 + 9.6 … 푎 24 푟 = 2 = = .4 푎1 60 푎 푆 = 1 푛 1 − 푟 60 푆 = 푛 1 − .4 푆푛 = 100 B. Applications Ex 1: Francisco designs a toy with a 푎 푆 = 1 rotary flywheel that rotates at a 푛 1 − 푟 maximum speed of 170 revolutions per minute. Suppose the flywheel is 170 푆 = operating at its maximum speed for one 푛 2 1 − minute and then the power supply to the 5 toy is turned off. Each subsequent minute thereafter, the flywheel rotates 푆푛 = 283.3333 two-fifths as many times as in the preceding minute. How many complete It makes 283 complete revolutions will the flywheel make before revolutions before it stops. coming to a stop? Ex 2: A tennis ball dropped from a height of 24 feet bounces .75% of the height from which it fell on each bounce. What is the vertical distance it travels before coming to rest? C. Writing Repeating Decimals as Fractions • To write a repeating decimal as a fraction, start by writing it as an infinite geometric sequence. Ex. 1: Write 0. 762 as a fraction 762 762 762 0. 762 = + + + ⋯ 1000 1,000,000 1,000,000,000 762 1 In this series, a = and r = 1 1000 1000 762 푎 762 254 푆 = 1 = 1000 = = 푛 1 1 − 푟 1 − 999 333 1000 Ex 2: Write 0.123123123… as a fraction using an Infinite Geometric Series. 123 123 123 0. 123 = + + + ⋯ 1000 1,000,000 1,000,000,000 123 1 In this series, a = and r = 1 1000 1000 123 푎 123 푆 = 1 = 1000 = 푛 1 1 − 푟 1 − 999 1000 1 Ex 3: Show that 12.33333… = 12 using a geometric series. 3 First, write the repeating part as a fraction. 3 3 3 0. 7 = + + + ⋯ 10 100 1,000 3 3 In this series, a = and r = 1 10 10 3 1 푎 3 1 So, 12.33333… = 12 푆 = 1 = 10 = = 3 푛 1 1 − 푟 1 − 9 3 10 II. Sigma Notation/Summation Notation In mathematics, the uppercase Greek letter sigma is often used to indicate a sum or series. This is called sigma notation. The variable n used with sigma notation is called the index of summation. Upper Limit (greatest value of n) Expression for the general Summation term symbol Lower Limit (least value of n) This is read “the summation from n = 1 to 3 of 5n + 1”. Substitute n = 1 into the equation and continue through n = 3. (5*1 + 1) + (5*2 +1) + (5*3 + 1) = 6+11+16 Expanded Form 33 A. Writing in Expanded Form and Evaluating Ex 1: Write the following in expanded form and evaluate: N = 10 1st = 1 Expanded form: Last = 10 -2 + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 a1= 1 - 3 = -2 a10 = 10 - 3 = 7 Evaluate: 25 Notice that since this is an arithmetic series, we could also use the formula for a finite arithmetic series to evaluate: n Sn (a1 an) 2 10 Sn (2 7) 5(5) 25 2 Ex 2: Find the number of terms, the first term and the last term. Then evaluate the series: N = 4 1st = 2 Last = 5 Note: this is NOT an arithmetic series. You can Expanded for: NOT use the formula; you have to manually crunch out 4+9+16+25 all the values. Evaluate: 54 B. n Factorial • As you have seen, not all sequences are arithmetic or geometric. Some important sequences are generated by products of consecutive integers. The product n(n – 2) … 3 * 2 * 1 is called n factorial and is symbolized n!. • As a rule, 0! = 1 • The table at right shows just how quickly the numbers can grown. Copy down the first 7 rows. You will need to recognize this pattern in a subsequent example. C. Writing a series in sigma notation • Ex 1: 102 + 104 + 106 + 108 + 110 + 112 n = 6 terms 1st term = 1 Rule: Hmmmm. Rule = 100 + 2n 4 16 64 256 • Ex 2:Write in sigma notation: − + − + 1 2 6 24 n = 4 terms 1st term = 1 Rule: Hmmmm. The numerator has powers of 4, and the signs rotate back and forth between positive and negative. The means the numerator should be (−1) 24푛 The denominator has factorials, so it should be n! So we have 4 (−1)24푛 푛! 푛=1 D. Applications • Ex 1: During a nine-hole charity golf • a. Since the sequence is match, one player presents the geometric, we can use the following proposition: The loser of the first hole will pay $1 to charity, formula for the nth term of a and each succeeding hole will be geometric sequence. worth twice as much as the hole n–1 an = a1r immediately preceding it. 4–1 • a. How much would a losing player an = 1(2) pay on the 4th hole? a = (2)3 • b. How much would a player lose if n he or she lost all nine holes? an = 8 • c. Represent the sum using sigma The loser would have to pay $8. notation. • b. We can use the formula for a finite • c. geometric series. Each time is 푎 − 푎 푟푛 doubled, we 푆 = 1 1 푛 1 − 푟 start with 1$, so • 9 total holes 20 = 1, 21 = 2, 2 1−1(2)9 2 = 4, etc. 푆 = 9 푛 1−2 푛−1 2 1−512 푆 = 푛=1 푛 1−2 푆푛 = 511 Start at the first hole The loser of all nine holes would have to pay $511. .
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