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4 th EDITION

College Algebra & Trigonometry and Precalculus. 4 th EDITION. 11.1. Sequences and Series. Sequences Series and Summation Notation Summation Properties. Sequences.

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4 th EDITION

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  1. College Algebra & Trigonometry and Precalculus 4th EDITION

  2. 11.1 Sequences and Series Sequences Series and Summation Notation Summation Properties

  3. Sequences A sequence is a function that computes an ordered list. For example, the average person in the United States uses 100 gallons of water each day. The function defined by (n) = 100n generates the terms of the sequence 100, 200, 300, 400, 500, 600, 700,…, when n = 1, 2, 3, 4, 5, 6, 7, …. This function represents the gallons of water used by the average person after n days.

  4. Sequences A second example of a sequence involves investing money. If $100 is deposited into a savings account paying 5% interest compounded annually, then the function defined by g(n) = 100(1.05)n calculates the account balance after n years. The terms of the sequence are g(1), g(2), g(3), g(4), g(5), g(6), g(7), …, and can be approximated as 105, 110.25, 115.76, 121.55, 127.63, 134.01, 140.71, . . . .

  5. Sequence A finite sequence is a function that has a set of natural numbers of the form {1, 2, 3, …, n} as its domain. An infinite sequence has the set of natural numbers as its domain.

  6. Sequences For example, the sequence of natural-number multiples of 2, 2, 4, 6, 8, 10, 12, 14, …, is infinite, but the sequence of days in June, 1, 2, 3, 4, …, 29, 30, is finite.

  7. Sequences Instead of using f (x) notation to indicate a sequence, it is customary to use an, where an = (n).The letter n is used instead of x as a reminder that n represents a natural number. The elements in the range of a sequence, called the terms of the sequence, are a1, a2, a3, …. The elements of both the domain and the range of a sequence are ordered. The first term is found by letting n = 1, the second term is found by letting n = 2, and so on. The general term, ornth term, of the sequence is an.

  8. Sequences These figures show graphs of (x) = 2x and an = 2n. Notice that (x) is a continuous function, and an is discontinuous. To graph an,we plot points of the form (n, 2n) for n = 1, 2, 3,….

  9. FINDING TERMS OF SEQUENCE Example 1 Write the first five terms for each sequence. a. Solution Replacing n in with 1, 2, 3, 4, and 5 gives

  10. FINDING TERMS OF SEQUENCE Example 1 Write the first five terms for each sequence. b. Solution Replace n in with 1, 2, 3, 4, and 5 to obtain

  11. FINDING TERMS OF SEQUENCE Example 1 Write the first five terms for each sequence. c. Solution Replacing n in we have

  12. Converge and Diverge If the terms of an infinite sequence get closer and closer to some real number, the sequence is said to be convergent and to converge to that real number. For example, the sequence defined by approaches 0 as n becomes large.

  13. Converge and Diverge Thus an, is a convergent sequence that converges to 0. A graph of this sequence for n = 1, 2, 3, …, 10 is shown here. The terms of an approach the horizontal axis.

  14. Converge and Diverge A sequence that does not converge to any number is divergent. The terms of the sequence are 1, 4, 9, 16, 25, 36, 49, 64, 81, …. This sequence is divergent because as n becomes large, the values of do not approach a fixed number; rather, they increase without bound.

  15. Recursive Definitions Some sequences are defined by a recursive definition, one in which each term after the first term or first few terms is defined as an expression involving the previous term or terms. On the other hand, the sequences in Example 1 were defined explicitly, with a formula for an that does not depend on a previous term.

  16. USING A RECURSIVE FORMULA Example 2 Find the first four terms of each sequence. a. Solution This is a recursive definition. We know a1 = 4. Since an = 2  an – 1 +1,

  17. USING A RECURSIVE FORMULA Example 2 Find the first four terms of each sequence. b. Solution This is a recursive definition. We know a1 = 2 and an = an – 1 + n – 1.

  18. MODELING INSECT POPULATION GROWTH Example 3 Frequently the population of a particular insect does not continue to grow indefinitely. Instead, its population grows rapidly at first, and then levels off because of competition for limited resources. In one study, the behavior of the winter moth was modeled with a sequence similar to the following, where an represents the population density in thousands per acre during year n.

  19. MODELING INSECT POPULATION GROWTH Example 3 a. Give the table of values for n = 1, 2, 3, …, 10 Solution Evaluate a1, a2, a3, …, a10. and

  20. MODELING INSECT POPULATION GROWTH Example 3 a. Give the table of values for n = 1, 2, 3, …, 10 Solution Approximate values for n = 1, 2, 3, …, 10 are shown in the table.

  21. MODELING INSECT POPULATION GROWTH Example 3 b. Graph the sequence. Describe what happens to the population density. Solution The graph of a sequence is a set of discrete points. Plot the points (1, 1), (2, 2.66), (3, 6.24), …,(10, 9.98), as shown here.

  22. MODELING INSECT POPULATION GROWTH Example 3 Solution At first, the insect population increases rapidly, and then oscillates about the line y = 9.7. The oscillations become smaller as n increases, indicating that the population density may stabilize near 9.7 thousand per acre

  23. Note In Example 3, the insect population stabilizes near the value k = 9.7 thousand. This value of k can be found by solving the quadratic equation k = 2.85k – .19k2. Why is this true?

  24. Series and Summation Notation Suppose a person has a starting salary of $30,000 and receives a $2000 raise each year. Then, 30,000 32,000 34,000 36,000 38,000 are terms of the sequence that describe this person’s salaries over a 5-year period.

  25. Series and Summation Notation The total earned is given by the finite series whose sum is $170,000. Any sequence can be used to define a series. Take a look at the following slide to see what this looks like.

  26. Series and Summation Notation Any sequence can be used to define a series. For example, the infinite sequence defines the terms of the infinite series

  27. Series and Summation Notation If a sequence has terms a1, a2, a3, …, then Sn is defined as the sum of the first n terms. That is, The sum of the terms of a sequence, called a series, is written using summation notation. The symbol , the Greek capital letter sigma, is used to indicate a sum.

  28. Series A finite series is an expression of the form and an infinite series is an expression of the form The letter iis called the index of summation.

  29. Caution Do not confuse this use of i with the use of i to represent the imaginary unit. Other letters, such as k and j, may be used for the index of summation.

  30. USING SUMMATION NOTATION Example 4 Evaluate the series Solution Write each of the six terms, then evaluate the sum.

  31. USING SUMMATION NOTATION Example 4 Evaluate the series Solution Write each of the six terms, then evaluate the sum.

  32. USING SUMMATION NOTATION WITH SUBSCRIPTS Example 5 Write the terms for each series. Evaluate each sum, if possible. a. Solution

  33. USING SUMMATION NOTATION WITH SUBSCRIPTS Example 5 b. Solution Use the order of operations. Substitute the given values for x1, x2, and x3.

  34. USING SUMMATION NOTATION WITH SUBSCRIPTS Example 5 c. Solution Simplify. Add.

  35. Summation Properties If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then for every positive integer n, (a) (b) (c) (d)

  36. Summation Properties To prove Property (a), expand the series to obtain where there are n terms of c, so the sum is nc. Property (c) also can be proved by first expanding the series: Commutative and associative properties.

  37. Summation Rules

  38. USING SUMMATION PROPERTIES Example 6 Use the summation properties to find each sum. a. Solution Property (a) with n = 40 and c = 5.

  39. USING SUMMATION PROPERTIES Example 6 Use the summation properties to find each sum. b. Solution Property (b) with c = 2 and ai = i Summation rules Simplify.

  40. USING SUMMATION PROPERTIES Example 6 c. Solution Property (d) with ai = 2i2 and bi = 3 Property (b) with c = 2 and ai = i2 Summation rules; Property (a) Simplify.

  41. USING SUMMATION PROPERTIES Example 7 Evaluate Solution Property (c) Property (b) Property (a)

  42. USING SUMMATION PROPERTIES Example 7 Evaluate Solution Property (c) Property (b)

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