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Infinite Sequences and Summation Notation

Infinite Sequences and Summation Notation. An arbitrary infinite sequence may be denoted as follows. a 1 , a 2 , a 3 , a 4 , ... a n ,. Each number a k is a term of the sequence. The sequence is ordered in the sense that there is a first term a 1 ,

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Infinite Sequences and Summation Notation

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  1. Infinite Sequences and Summation Notation

  2. An arbitrary infinite sequence may be denoted as follows. a1, a2, a3, a4, ... an, ... Each number ak is a term of the sequence.

  3. The sequence is ordered in the sense that there is a first term a1, a second term a2, a thirty third term a33, and if n represents an arbitrary positive integer, and nth term an.

  4. For our work we will restrict the domain to the set of positive integers and define an infinite sequence as: An infinite sequence is a function whose domain is the set of positive integers.

  5. For our work the range of an infinite sequence will be the set of real numbers. With these definitions each positive integer n in the domain of the sequence corresponds to a real number ƒ(n) = an.

  6. c. a. (-1)n+1 n n+1 n2 3n-1 Example 1. Find the first four terms and the tenth term of each sequence. b. {2 + (0.1)n } d. {4} To find the above replace n with 1, 2, 3, 4, and then 10.

  7. = a. n n+1 1 1+1 1 2 2 2+1 3 3+1 10 11 4 4+1 4 5 10 10+1 2 3 3 4 = = = = Example 1. Find the first four terms and the tenth term of each sequence.

  8. Example 1. Find the first four terms and the tenth term of each sequence. b. {2 + (0.1)n } 2 + (0.1)1 = 2.1 2 + (0.1)2 = 2.01 2 + (0.1)3 = 2.001 2 + (0.1)4 = 2.0001 2 + (0.1)10 = 2.0000000001

  9. c. (-1)n+1 = = (-1)1+1 (-1)2+1 n2 3n-1 12 3•1-1 1 2 22 3•2-1 -4 5 Example 1. Find the first four terms and the tenth term of each sequence.

  10. c. (-1)n+1 = = (-1)3+1 (-1)4+1 n2 3n-1 32 3•3-1 9 8 42 3•4-1 -16 11 Example 1. Find the first four terms and the tenth term of each sequence.

  11. c. (-1)n+1 = (-1)10+1 n2 3n-1 102 3•10-1 -100 29 Example 1. Find the first four terms and the tenth term of each sequence.

  12. Example 1. Find the first four terms and the tenth term of each sequence. d. {4 } 4 = 4 4 = 4 4 = 4 4 = 4 4 = 4

  13. Advanced Algebra and Trigonometry Assignment 04-01-03 NBP#40 Page 671 Exercises 1 - 15odd Copy each problem. Show your work. Underline your answer.

  14. Advanced Algebra and Trigonometry Notes 04-02-03 NBP#41 Infinite Sequences and Summation Notation Pages 664 - 670

  15. Definition: Recursive sequence The first term a1 is stated along with a rule for obtaining any term ak+1 from the preceding term ak whenever k > 1.

  16. Example 2. Find the first four terms and the nth term of the infinite sequence defined recursively as: ak+1 = 2ak for k > 1. a1 = 3 a2 = 2•3 a3 = 2•2•3 an = 2n-1•3 a4 = 2•2•2•3

  17. We sometimes need to find the sum of many terms in an infinite sequence. To express these sums easily we use summation notation. Given an infinite sequence a1, a2, a3, ... , an, ...

  18. m k=1 The symbol  ak represents the sum of the first m terms. m k=1  ak = a1 + a2 + a3 + ... + am Given an infinite sequence a1, a2, a3, ... , an, ...

  19. m k=1 The symbol  ak represents the sum of the first m terms. m k=1  ak = a1 + a2 + a3 + ... + am The symbol  represents a sum, ak represents the kth term.

  20. m k=1  ak = a1 + a2 + a3 + ... + am The symbol  represents a sum, ak represents the kth term. The letter k is the index of summation, or the summation variable. 1 and m indicate the smallest and largest values of the summation variable.

  21. 4 k=1 ∑ k2(k-3) 4 k=1 ∑ k2(k-3) Example 3. Evaluate a sum. = 12(1-3) + 22(2-3) + 42(4-3) + 32(3-3) = (-2) + (-4) + 0 + 16 = 10

  22. Advanced Algebra and Trigonometry Assignment 04-01-03 NBP#42 Page 671 Exercises 17 - 31odd Copy each problem. Show your work. Underline your answer.

  23. Advanced Algebra and Trigonometry Notes 04-03-03 NBP#43 Infinite Sequences and Summation Notation Pages 664 - 670

  24. We are talking about Sn. For example given the infinite sequence a1, a2, a3, ... , an, ... S1 = a1 S2 = a1 + a2 S3 = a1 + a2 + a3 S4 = a1 + a2 + a3 + a4

  25. n k=1 In general Sn = ∑ ak = a1+a2+...+an We may also write S1 = a1 S2 = S1 + a2 S3 = S2 + a3 S4 = S3 + a4 Sn = Sn-1 + an

  26. n k=1 In general Sn = ∑ ak = a1+a2+...+an We may also write S1 = a1 S2 = S1 + a2 S3 = S2 + a3 S4 = S3 + a4 Sn = Sn-1 + an

  27. The real number Sn is called the nth partial sum of the infinite sequence a1, a2, a3, ... an, ... The sequence S1, S2, S3, ... Sn, ... is called a sequence of partial sums.

  28. Example 4. Finding the terms of a sequence of partial sums. Find the first four terms and the nth term of the sequence of partial sums associated with the sequence 1, 2, 3, ..., n, ... of positive integers. The first four terms are: S1 = a1 = 1

  29. Example 4. Finding the terms of a sequence of partial sums. The first four terms are: S1 = a1 = 1 S2 = S1 +a2 = 1 + 2 = 3 S3 = S2 +a3 = 3 + 3 = 1+2+3 = 6 S4 = S3 +a4 = 6 + 4 = 1+2+3+4 = 10

  30. Example 4. Finding the terms of a sequence of partial sums. The nth partial sum Sn can be written in either of the following forms. Sn = 1+2+3+...+(n-2)+(n-1)+n Sn = n+(n-1)+(n-2)+...+3+2+1

  31. Example 4. Finding the terms of a sequence of partial sums. When I add these two equations Sn = 1+2+3+...+(n-2)+(n-1)+n Sn = n+(n-1)+(n-2)+...+3+2+1 2Sn = (n+1)+(n+1)+...+(n+1)

  32. n(n+1) 2 Example 4. Finding the terms of a sequence of partial sums. 2Sn = (n+1)+(n+1)+...+(n+1) Since the term (n+1) appear n times on the right side of the equations we get. 2Sn = n(n+1) or Sn =

  33. n k=1 ∑ ak = Sum of a constant If ak is the same for every positive integer k, say ak = c for a real number c, then. a1 + a2 + a3 + ... + an = c + c + c + ... + c = nc

  34. n k=m n k=1 m-1 k=1 ∑ c = ∑ c - ∑ c Sum of a constant Part 2 If ak is the same for every positive integer k, say ak = c for a real number c, then. = (n - m + 1)c = nc - (m-1)c

  35. 4 k=1 10 k=1 8 k=3 ∑ 7 = ∑ π = ∑ 9 = Sum of a constant - Illustrations 4 • 7 = 28 10 • π = 10π (8 - 3 + 1)9 = (6)9 = 54

  36. 20 k=10 ∑ 5 = Sum of a constant - Illustrations = (11)9 = 55 (20 - 10 + 1)5

  37. n k=0 Sn = ∑ ak = a0+a1+a2+...+an If the first term of an infinite sequence is a0, as in a0, a1, a2, ... , an, ... which is the sum of the first n+1 terms of the sequence.

  38. If the summation variable does not appear in the term ak, then the entire term must be considered a constant. n j=1 For example ∑ ak = n•ak since j does not appear in the term ak.

  39. n k=1 n k=1 n k=1 ∑ (ak + bk) ∑ (ak ∑ bk) Theorem on sums. If a1, a2, … , an, … and b1, b2, … , bn, … are infinite sequences, for every positive integer n, + =

  40. n k=1 n k=1 n k=1 ∑ (ak - bk) ∑ (ak ∑ bk) Theorem on sums. If a1, a2, … , an, … and b1, b2, … , bn, … are infinite sequences, for every positive integer n, - =

  41. n k=1 ∑ cak Theorem on sums. If a1, a2, … , an, … and b1, b2, … , bn, … are infinite sequences, for every positive integer n, n k=1 c(∑ ak) =

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