Math Example Set 14A Section 10.1 Sequences Section 10.2 Summing an Infinite Series


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1 Math Example Set 14A Section 10.1 Sequences Introduction to Geometric Sequence and Series 1. A ball is projected from the ground to a height of 10 feet and allowed to freely fall and rebound when it hits the ground. If the ball rebounds to 80% of the height it fell from and is allowed to continue its motion indefinitely answer the questions below. a. List the height attained by the ball just before it hits the ground the nth time for n = 1, 2, 3, and 4. Draw a picture to explain how you arrive at your answer. Identify any pattern you observed and write down he maximum height h n attained by the ball before it hits the ground the nth time. The list of numbers {h n } is called a sequence. What is the limit of this sequence? b. Show that the sequence {h n } in Q1 is geometric. What is its first term and common ratio? c. Write out as a sum, the total vertical distance travelled by the ball when its hits the floor the first time, second time, third times and fourth time. If the bouncing persists, write down using summation notation the total distance travelled by the ball. We called this a series. In particular, since it is a sum of a geometric sequence we call it a geometric series. Is the total vertical distance travelled by the ball finite or infinite? Make a guess? Definition (Geometric Sequence): A sequence {a n } is said to be geometric if the ratio of its consecutive terms a n+1 a n is a. We call such a sequence a geometric sequence with first term a 1 = c and common ratio r. The nth term a n of the sequence is given by. Definition (Geometric Series): We called an infinite sum (series) of the form c + cr + cr 2 + cr cr n 1 + a geometric series with first term c and common ratio r. General Sequences and their limits 2. Find the general term of the following sequences, assuming that the pattern of the first few terms continues. Show whether it is geometric. If it is geometric, state its first term and common ratio. Find also the limit of each sequence. a. { 4 5, 8 25, 16 } 125, , b. { 1 2, 2 3, 3 4, 4 } 5, c. c n+1 = (n + 1)c n for n 1 and c 1 = 1 1
2 Partial Sum of a Geometric Series We want to make sense of the sum of a geometric series. But that would mean we need to understand adding infinitely many numbers. To understand this we first find a formula for finite partial sums of a geometric series. The sum formula for the first N terms of a geometric series: c + cr + cr 2 + cr cr N 1 = c ( 1 r N) 1 r This is also called the Nth partial sum of a geoemtric series with first term c and common ratio r Proof: Consider first the finite sum S N = c + cr + cr cr N 1 S N = c + cr + cr 2 + cr cr N 1 rs N = cr + cr 2 + cr cr N 1 + cr N S N rs N =? So (1 r)s N? = S N? = 3. Find the sum of the first 20 terms of the following geometric series. 3a b. n=3 2 2n 3 n 2
3 Math Example Set 14B Definition (Geometric Series): We called an infinite sum (series) of the form c + cr + cr 2 + cr cr n 1 + a geometric series with first term c and common ratio r. Sum formula for Geometric Series: Consider a geometric series with first term c and common ratio r. If r < 1 then the geometric series is convergent. Moreover, c + cr + cr 2 + cr cr n 1 + = If r > 1, then the geometric series is divergent. c 1 r. Proof: Consider the finite sum S N = c + cr + cr cr N 1 = c ( 1 r N) 1 r If r < 1, lim n S N? = If r > 1, lim N S N? = Summary: The sum formula for the first N terms of a geometric series: c + cr + cr 2 + cr cr N 1 = The sum formula for the geometric series: c + cr + cr 2 + cr 3 + = 1. For the following geometric series find its infinite sum if it exists. Explain why the sum exists or not. 1a b. Geometric Series and its Applications n=3 2. A ball is projected from the ground to a height of 10 feet and allowed to freely fall and rebound when it hits the ground. Suppose the ball rebounds to 80% of the height it fell from and is allowed to continue its motion indefinitely. Find the total vertical distance travelled by the ball. 3. Rewrite each of the following repeated decimals as a fraction. 3a = 0.999?= 3b = ?= 2 2n 3 n 3
4 4. A drug is designed so that 60% remains in the body at the end of each 24 hour period (one day). If 30 mg of the drug is given daily to a patient find (A) the amount of drug in the body after 10 days before the next dose is given, and (B) the approximate amount of drug in the body after a very long time assuming measurement is done before the next dose is given. 5. Consider a population of a single cell organism that changes in size only through birth and death. Suppose the birth rate (fertility rate) is 0.2 per year and death rate is 0.3 per year. Then if x is the population size, the number of birth is 0.2x and the number of death is 0.3x for the year. If the current (initial) population is 100 thousand, and x n denote the population after n years, write down a relation between x n and x n 1 (We call such relations an iterative formula or recursive relation for x n ). Find a formula for x n in terms of n. Is the population growing or declining? 4
5 Geometric Series and its Applications Math Example Set 14C 1a. (Q5 in Example Set 14B) Consider a population of a single cell organism that changes in size only through birth and death. Suppose the birth rate (fertility rate) is 0.2 per year and death rate is 0.3 per year. Then if x is the population size, the number of birth is 0.2x and the number of death is 0.3x for the year. If the current (initial) population is 100 thousand, and x n denote the population after n years, write down a relation between x n and x n 1 (We call such relations an iterative formula or recursive relation for x n ). Find a formula for x n in terms of n. Is the population growing or declining? 1b. If net immigration rate into the population above is 800 hundred (0.8 thousand) a year. Find an new iterative formula for x n. Find as time progresses, how large can the population get? Summing Telescopic Series 2 2. Consider the series. Answer the following questions: k 2 + 7k + 12 k=0 a. Apply partial fraction decomposition to find S 49, the sum of its first 49 terms (49th partial sum of the given series). b. Find S N the Nth partial sum of the given series (Nth partial sum of the given series). c. Is the given series convergent? Find the value of the given series. Watch the pencast of the following example in Sakai (Video & Notes). ( n ) 3a. Find the sum of the first 51 terms in the series + 1 n. 3b. Is the series ( n ) + 1 n convergent? n=6 Definition 1. (Partial Sum of an Infinite Series) Consider the infinite series sum of the series is the finite sum: S N = N a n. Definition 2. (Convergence of an Infinite Series) An infinite series n=6 a n. The Nth partial a n is said to be convergent if its sequence {S N } N=1 of Nth partial sums is convergent. More specifically, we mean that: (a) If lim S N = S then we say that the series coverges to S and write a n = S (the sum of the N infinite series). (b) If lim S N does not exist then we say that the series diverges and the sum of the infinite series N a n does not exist. 5
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