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Sequences and Series From Simple Patterns to Elegant and Profound Mathematics

Sequences and Series From Simple Patterns to Elegant and Profound Mathematics. David W. Stephens The Bryn Mawr School Baltimore, Maryland PCTM – 28 October 2005. Contact Information. Email: stephensd@brynmawrschool.org The post office mailing address is: David W. Stephens

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Sequences and Series From Simple Patterns to Elegant and Profound Mathematics

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  1. Sequences and SeriesFrom Simple Patterns to Elegant and Profound Mathematics David W. Stephens The Bryn Mawr School Baltimore, Maryland PCTM – 28 October 2005

  2. Contact Information Email: stephensd@brynmawrschool.org The post office mailing address is: David W. Stephens 109 W. Melrose Avenue Baltimore, MD 21210 410-323-8800 The PowerPoint slides will be available on my school website: http://207.239.98.140/UpperSchool/math/stephensd/StephensFirstPage.htm , listed under “PCTM October 2005”

  3. Teaching Sequences and Series We will look at some ideas for teaching sequences and series as well as some applications in mathematics classes at THREE different levels: I. Early (Algebra 1, Algebra 2, and Geometry) II. Intermediate (Advanced Algebra, Precalculus) III. Advanced (AP Calculus, esp. BC Calculus)

  4. Teaching Sequences and Series Many of the topics and examples used today will not be new to you, but I want you to consider thinking of them …and talking about them with students … as sequences and series. It can be a good way for them to think about these diverse topics as bring linked mathematically. Just as functions link a lot of what we teach, the patterns of sequences and series can tie these ideas together for better comprehension.

  5. Early Sequences and Series(Algebra 1, Algebra 2, and Geometry) 1. Looking for patterns 2. Identifying kinds of sequences 3. Describing patterns in sequences 4. Using variables 5. Summation notation 6. Strategies for summing 7. Applications with geometry ideas 8. Graphing patterns 9. Data analysis & functions as sequences

  6. Intermediate Sequences and Series(Advanced Algebra, Precalculus) 1. More geometric sequences and exponential functions 2. Infinite series 3. Convergence and divergence 4. Informal limits 5. More advanced data analysis (“straightening data”) 6. Applications (compound interest, astronomy, chemistry, biology, economics, periodic motion or repeating phenomena)

  7. Advanced Sequences and Series(AP Calculus, esp. BC Calculus) 1. Newton’s method for locating roots 2. Riemann sums 3. Trapezoid rule and Simpson’s rule 4. Euler’s method for differential equations 5. Power series (Maclaurin and Taylor series polynomials) 6. Convergence tests for series

  8. Early Topics(Algebra 1 , Algebra 2, and Geometry) I sometimes have begun my Algebra 2 classes in September with this topic because… a) New students to the school (and the class) do not feel “new”. b) I can use algebraic language. c) I can review linear functions in a new context. d) I can sneak in some review which does not feel like review! Early Sequences and Series

  9. Activity 1 Find the next three numbers in these sequences: A) 6, 9, 13, 18, 24, … B) 12, 17, 13, 16, 14, 15, 15, … C) 5, 10, 20, 40, … D) 7, -21, 63, -189, … E) 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, 0, 1, 2, 3, … Early Sequences and Series

  10. Activity 2 Students build their own sequences, and they challenge their classmates to guess the next few entries. This can be a neat homework assignment. (It can be extended to later activities where they have to code their sequence patterns with variables, too.) Early Sequences and Series

  11. Activity 3 Describe the pattern in words: A) 7, 5, 3, 1, -1, -3, … B) 70, 68, 64, 56, 40, 32, 28, 26, 25, … C) D) 12, 13, 14, 15, 16, 15, 14, 13, 12, 13, 14, 15, ... E) 1, 2, 3, 4, 4,3, 2, 1, 3, 5, 7, 7 , 5, 3, 1, 12, 23, 34, 23, … F) 4, 9, 32, 50, 53, 54, 54, 54, 54, 54.1, 54.13, 54.135, 54.1356 , ... Early Sequences and Series

  12. Activity 4 Learn to code the pattern with variables: A) 9, 13, 17, 21, 25, 29 , … Let a0 = 9 an = an-1+ 4 or an = a0+ (n-1)d (Some texts use tn, where t = term, instead of an) It could also be coded that a6 = 9 and then a7 = 13, if you decided to start the count at item #6 Early Sequences and Series

  13. Activity 4 (continued) B) 3, 6, 12, 24, 48, … Let a0 = 3 an = a0r n-1 = (first term)(ratioterm# - 1) ( or let t0 = 3 tn = a0r n-1) Early Sequences and Series

  14. Activity 5 Introduction to Fibonacci sequences A) 1, 1, 2, 3, 5, 8, 13, 21, 34 B) 2, 5, 7, 12, 19, 31, 50 an = an-1+ an-2 (recursively defined functions) Early Sequences and Series

  15. Activity 6 Series and Summation Notation 1. a) sequence 1 , 3 , 5 , 7 , 9 , 11 b) series 1 + 3 + 5 + 7 + 9 + 11 c) series notation or or Early Sequences and Series

  16. Activity 6 (continued) 2. a) sequence 2, 6, 18, 54 b) series 2 + 6 + 18 + 54 c) series notation 3. a) sequence 4, -2, 1, , … b) series 4 – 2 + 1 - + - … c) series notation Early Sequences and Series

  17. Activity 7 Interleaved and other creative sequences Find the next three terms, and describe the two sequences that are interleaved. A) 1, 3, 4, 9, 7, 27, 10, 81, 13, 243 , … 1, 3, 4, 9, 7, 27, 10, 81, 13, 243 , … B) 5, 1, 7, 4, 9, 7, 11, 10, 13, 13, 5, 1, 7, 4, 9, 7, 11, 10, 13, 13, … Early Sequences and Series

  18. Introduction to Series How to add up an arithmetic series efficiently: Example: sn = 6 + 9 + 12 + 15 + 18 + … + 219 • Add the first and last terms, the second and the second to the last, etc. What do you notice? • How many pairs are there? • What if there are an odd number of terms to add? Sn = Early Sequences and Series

  19. Introduction to Series How to add up a geometric series efficiently: Example: sn = 5 + 10 + 20 + 40 + 80 + 160 + 320 = a0 + a0r + a0r2 + a0r3+ … + a0rn-1 rsn = a0r + a0r2 + a0r3 + a0r4 + … +a0rn Then sn – rsn = a0 – a0rn This provides us with the usual formula for a geometric series: sn = Early Sequences and Series

  20. Activity 8 For the series, sn = 5 + 10 + 20 + 40 + … + 320 + … calculate s28 Early Sequences and Series

  21. Activity 9 Early Sequences and Series

  22. Activity 10 Early Sequences and Series

  23. Data Analysis Building Functions from Data Early Sequences and Series

  24. Data Analysis Building Functions from Data Early Sequences and Series

  25. Data Analysis Building Functions from Data Early Sequences and Series

  26. GeometryAngles of Polygons What is the general formula for the sum of the interior angles of a polygon with n sides? (n, measures of interior angles) : (3, 180) , (4, 360), (5 , 540) , (6 , 720) , … (n , 180(n-2)) Early Sequences and Series

  27. GeometryA Modeling Application Handshake Problem: If n people shakes hands with everyone else at a meeting, how many handshakes occur? 1. Visualize this as a geometry problem. 2. Consider a simpler version with just a few number of people. 3. Generalize the data, and consider the data as sequence. Early Sequences and Series

  28. GeometryA Modeling Application Handshake Problem: Early Sequences and Series

  29. GeometryA Modeling Application n = number of people h(n) = number of handshakes n 1 2 3 4 5 6 7 h(n) 0 1 3 6 10 15 21

  30. GeometryA Derivation of Find the perimeter of a sequence of regular polygons which are inscribed in a unit circle, and emphasize that the sequence of results is important to watch. s = length of one side of the polygon p = perimeter of the polygon Early Sequences and Series

  31. GeometryA Derivation of s = length of one side p = perimeter of inscribed polygon s = p = 4 = 5.657 s = p = 3= 5.196 Early Sequences and Series

  32. GeometryA Derivation of s = 1 p = 6 s = 2 sin(36) = 1.176 p = 5(1.176) = 5.878 Early Sequences and Series

  33. GeometryA Derivation of In general, the length of one-half of a side of an inscribed regular polygon is So a side measures and the perimeter of the polygon measures Since p  2 , then can be calculated. Early Sequences and Series

  34. GeometryA Derivation of The central angle for each side is Each half-side has length equal to the sine of one-half the central angle. Early Sequences and Series

  35. GeometryA Derivation of Here are the perimeters of the polygons from the TI-83 as a list (L2) Note: Ignore L3. Early Sequences and Series

  36. Intermediate Sequences and Series(Advanced Algebra, Precalculus) 1. More geometric sequences and exponential functions 2. Infinite series 3. Convergence and divergence 4. Informal limits 5. More advanced data analysis (“straightening data”) 6. Applications (compound interest, astronomy, chemistry, biology, economics, periodic motion or repeating phenomena) Intermediate Sequences and Series

  37. Data AnalysisBuilding Functions from Data Example 4: x 1 2 3 4 5 6 7 8 y 3 9 27 81 243 729 2187 6561 x is an arithmetic sequence y is a geometric sequence This is sometimes called an “add-multiply” property So y = f(x) is EXPONENTIAL What is the actual function? Ans: f(x) = 3x ( where r = 3 in the geometric sequence) Intermediate Sequences and Series

  38. Data AnalysisBuilding Functions from Data Example 5: x 1 2 3 4 5 6 7 8 y 5 11 29 83 245 731 2189 6563 y 3 9 27 81 243 729 2187 6561 x is an arithmetic sequence y is not exactly a geometric sequence But if the sequence of y-values is compared with the last set of y’s, then we see that this sequence is 2 more than a geometric sequence. So y = 3x + 2 Intermediate Sequences and Series

  39. Data AnalysisBuilding Functions from Data Example 6: x 1 4 7 10 13 y 6 48 384 3072 24,576 x is an arithmetic sequence y is not exactly a geometric sequence Since the two sequences have the “add-multiply” property, then y is a geometric sequence, and it is exponential. Notice that the x’s do not have to be consecutive. We have to find the “r” value as if we are calculating geometric means Intermediate Sequences and Series

  40. Data AnalysisBuilding Functions from Data Example 7: x 1 4 7 10 13 y 6 48 384 3072 24,576 a1 = 6 and a4 = 48, and we need to fill in the sequence so that we know the y-values for terms 2 and 3. Since the desired sequence is geometric, we need to know what to multiply a1 by repeatedly three times to get 48. This suggests that r*r*r= 48/6. So r = = 2 , and y = 3 * 2x Intermediate Sequences and Series

  41. Data AnalysisBuilding Functions from Data Example 7: x 1 2 3 4 5 6 7 8 9 10 11 y 6 12 24 48 96 192 384 768 1536 3072 6144 r = = 2 , and y = 3 * 2x Intermediate Sequences and Series

  42. An Historical Diversion Let’s take a look at thepairing of an arithmetic and a geometric sequence. n 1 2 3 4 5 6 an 2 4 8 16 32 64 Let’s suppose that we wanted intermediate terms: n 1 3/2 2 5/2 3 7/2 4 9/2 5 6 an 2 4 8 16 32 64 Intermediate Sequences and Series

  43. An Historical Diversion n 1 3/2 2 5/2 3 7/2 4 9/2 5 an 2 4 8 16 32 Thinking about an as a geometric sequence, we need a geometric mean to fill in the missing terms. Our desired multiplier, r, is . an 2 4 8 16 32 an 2 2 3/2 4 2 5/2 8 2 7/2 16 2 9/2 32 Intermediate Sequences and Series

  44. An Historical Diversion So when we write an = 2n , then the sequence, n, becomes the exponents, or the logarithms, for the geometric sequence. This is part of the history of Henry Briggs, John Napier, Jobst Burgi, John Wallis, and Johann Bernoulli from 1620 to 1749 in the development of logarithms. Intermediate Sequences and Series

  45. Function Transformations using Sequences If functions are considered as lists of data, and one function is a transformation of another one, then the alterations to the sequence of function values is the key to decoding the transformation. We want to write g(x) as a transformation of f(x), so g(x) = f(x – 3)

  46. Function Transformations using Sequences Preliminary questions: A. When a transformation such as f(x + a) is used, what happens to the y values? B. When a transformation such as f(x) + a is used, what happens to the y values? C. When a transformation such as a*f(x) is used, what happens to the y values? g(x) = f(x – 5) + 1

  47. Function Transformations using Sequences g(x) = 2f(x – 2)

  48. Infinite Sequences, Series and Convergence There are some really good opportunities to lead students to important conclusions, as well as to challenge their intuition with some sophisticated ideas … with infinite sequences and series. We can extend their numerical sense as well as exploiting their graphical skills to help generate conclusions. Intermediate Sequences and Series

  49. Infinite Sequences, Series and Convergence Suppose an: 1, 3, 5, 7, 9, … Where does an go as n gets large? Suppose bn: 1, 1.01, 1.02 , 1.03 , 1.04 , … Where does bn go as n gets large? Suppose cn: 1, 2, 4, 8, 16, … Where does cn go as n gets large? Intermediate Sequences and Series

  50. Infinite Sequences, Series and Convergence Suppose dn: Where does dn go as n gets large? Since this is the ratio of two sequences, each of which approaches infinity, explain your answer to this question. Intermediate Sequences and Series

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