1. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §10.1 Inf Series

2. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.4 More Differential Equation Applications  Any QUESTIONS About HomeWork §9.4 → HW-16 9.4

3. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 3 Bruce Mayer, PE Chabot College Mathematics §10.1 Learning Goals  Determine convergence or divergence of an infinite series  Examine and use geometric series

4. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 4 Bruce Mayer, PE Chabot College Mathematics Infinite SEQUENCE  An infinite sequence is a function which Has the domain of all NATURAL Numbers A constant Mathematical Relationship between adjacent elements a 1, a 2, a 3, a 4,..., a n,... Elements  The 1 st 3 elements of the sequence a n = 2n 2 a 1 = 2(1) 2 = 2 a 2 = 2(2) 2 = 8 a 3 = 2(3) 2 = 18 Finite Sequence

5. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 5 Bruce Mayer, PE Chabot College Mathematics Arithmetic vs. Geometric Seq  ARITHMETIC Sequence → Repeatedly ADD a number, d (a difference), to some initial value, a  GEOMETRIC Sequence → start with a number a and repeatedly MULTIPLY by a fixed nonzero constant value, r ( a ratio)

6. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 6 Bruce Mayer, PE Chabot College Mathematics GeoMetric Sequence  A sequence is GEOMETRIC if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512,... Geometric Sequence The common ratio, r, is 4

7. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 7 Bruce Mayer, PE Chabot College Mathematics GeoMetric Sequence: “ n th ” Term  The n th term of a geometric sequence has the form: a n = a 1 r n−1 where r is the common ratio of consecutive terms of the sequence  Example The n th term is 15(5 n-1 ) a 1 = 15 a 2 = 15(5) a 3 = 15(5 2 ) a 4 = 15(5 3 ) 15, 75, 375, 1875,...

8. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  GeoMetric Seq  Determine a 1, r,and the n th term for the GeoMetric Sequence  Recognize: a 1, = 1, and r = ⅓  The n th term is: a n = (⅓) n–1

9. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 9 Bruce Mayer, PE Chabot College Mathematics Summation Notation  Represent the first n terms of a sequence by summation notation.  Example index of summation upper limit of summation lower limit of summation

10. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 10 Bruce Mayer, PE Chabot College Mathematics Finite Sum for GeoMetric Sequence  The Sum of a Finite Geometric Sequence Given By 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a 1 = 5

11. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 11 Bruce Mayer, PE Chabot College Mathematics INFinite Sum for GeoMetric Seq  The sum of the terms of an INfinite geometric sequence is called a Geometric Series  If |r| < 1, then the infinite geometric series has the Sum:  If |r| ≥ 1, then the infinite geometric series Does NOT have a Sum (it Diverges) a 1 + a 1 r + a 1 r 2 + a 1 r 3 +... + a 1 r n-1 +...

12. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Infinite Series  Find the sum of  Recognize:  Thus the Series Sum:

13. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 13 Bruce Mayer, PE Chabot College Mathematics nth Partial Sum of a Series  The General form of an Infinite Series  Then a Finite Fragment of the Sum is called the n th Partial Sum → Where n is simply any Natural Number (say 537)

14. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 14 Bruce Mayer, PE Chabot College Mathematics Convergence vs. Divergence  An Infinite series with n th Partial Sum  CONVERGES to sum S if S is a Finite Number such that  In this Case  The Series is said to DIVERGE when i.e., the Limit Does Not Exist

15. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Divergence  This Series DIVERGES  Note that the quantity {1+3n} increases without bound  Then the partial Sum: Always Increase as K increases

16. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Convergence  It is known that the following Leibniz series converges to the value π /4 as n →∞ for the Partial Sum:  This Convergence is difficult to Prove, so Check numerically for n: 1→200

17. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 17 Bruce Mayer, PE Chabot College Mathematics MATLAB Code for Leibniz % Bruce Mayer, PE % ENGR25 * 12Apr14 % file = MTE_Leibinz_Series_1404.m % clear; clc; clf; % N = 100 % the Number terms in the Sum N+1 for n = 1:N k = 0:n; S(n) = sum((-1).^k.*(1./(2*k+1))); end % Calc DIFFERENCE compare to pi/4 % % The y = PI Lines zxh = [0 N]; zyh = [pi/4 pi/4]; % whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green axes; set(gca,'FontSize',12); plot((1:N),S,'b', 'LineWidth',1.5), grid,... xlabel('\fontsize{14}n'), ylabel('\fontsize{14}Sum & \pi/4'),... title(['\fontsize{16}MTH16 Leibniz Series',]) hold on plot(zxh,zyh, 'g', 'LineWidth', 2) hold off

18. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 18 Bruce Mayer, PE Chabot College Mathematics Sum & Multiple Rules

19. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Use Sum & Mult Rules  Assume that this Series Converges to 4:  Use this information to find the value of  SOLUTION  Using properties of convergent infinite series, find →

20. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Negative Advertising  A Social Science study suggests that Negative political ads work, but only over short periods of time. Assume that a Negative ad influences the vote of 500 voters, but that influence decays at an instantaneous rate of 40% per day.  Find the number of influenced voters (a) as a partial sum if Negative ads are run each day for a week and (b) if the ads were continued at a daily rate indefinitely.

21. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Negative Advertising  SOLUTION: a)Each ad influences 500 voters initially, and then drops off precipitously: only a fraction of e −0.40t total voters remain influenced after t days. Thus the partial sum over a week of advertising: Thus The ads influence about 955 voters during the week.

22. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Negative Advertising b)The infinite sum calculates the effect of continuing the ads indefinitely  So The ads influence about 1017 voters if continued indefinitely - less than 100 additional votes compared to running the ads for only one week.

23. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 23 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §10.1 P49 Follow the Bouncing Ball

24. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 24 Bruce Mayer, PE Chabot College Mathematics All Done for Today Series: Arithmetic Geometric

25. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 25 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

26. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 26 Bruce Mayer, PE Chabot College Mathematics

27. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 27 Bruce Mayer, PE Chabot College Mathematics Bouncing Ball

28. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 28 Bruce Mayer, PE Chabot College Mathematics

29. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 29 Bruce Mayer, PE Chabot College Mathematics

30. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 30 Bruce Mayer, PE Chabot College Mathematics

31. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 31 Bruce Mayer, PE Chabot College Mathematics

32. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 32 Bruce Mayer, PE Chabot College Mathematics

33. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 33 Bruce Mayer, PE Chabot College Mathematics

34. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 34 Bruce Mayer, PE Chabot College Mathematics