Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §10.1Inf Series Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 9.4 Review § • Any QUESTIONS About • §9.4 More Differential Equation Applications • Any QUESTIONS About HomeWork • §9.4 → HW-16

3. §10.1 Learning Goals • Determine convergence or divergence of an infinite series • Examine and use geometric series

4. Infinite SEQUENCE • An infinite sequence is a function which • Has the domain of all NATURAL Numbers • A constant Mathematical Relationship between adjacent elements • The 1st 3 elements of the sequence an = 2n2 a1, a2, a3, a4, . . . , an, . . . a1= 2(1)2 = 2 Elements FiniteSequence a2= 2(2)2 = 8 a3= 2(3)2 = 18

5. 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. 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. a2 = 15(5) a3 = 15(52) a4 = 15(53) GeoMetric Sequence: “nth” Term • The nthterm of a geometric sequence has the form: an = a1rn−1 • where r is the common ratio of consecutive terms of the sequence • Example • The nth term is 15(5n-1) a1 = 15 15, 75, 375, 1875, . . .

8. Example  GeoMetricSeq • Determine a1, r ,and the nth term for the GeoMetric Sequence • Recognize: a1, = 1, and r = ⅓ • The nth term is: an = (⅓)n–1

9. Summation Notation • Represent the first n terms of a sequence by summation notation. • Example upper limit of summation lower limit of summation index of summation

10. Finite Sum for GeoMetric Sequence • The Sum of a Finite Geometric Sequence Given By 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a1 = 5

11. INFinite Sum for GeoMetricSeq • 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) a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .

12. Example  Infinite Series • Find the sum of • Recognize: • Thus the Series Sum:

13. nth Partial Sum of a Series • The General form of an Infinite Series • Then a Finite Fragment of the Sum is called the nth Partial Sum → • Where n is simply any Natural Number (say 537)

14. Convergence vs. Divergence • An Infinite series with nth 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. Example  Divergence • This Series DIVERGES • Note that the quantity {1+3n} increases without bound • Then the partial Sum: Always Increase as K increases

16. 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. 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. Sum & Multiple Rules

19. 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. 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. Example  Negative Advertising • SOLUTION: • Each ad influences 500 voters initially, and then drops off precipitously: only a fraction of e−0.40ttotal 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. Example  Negative Advertising • 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. WhiteBoard Work • Problems From §10.1 • P49 Follow the Bouncing Ball

24. All Done for Today Series: ArithmeticGeometric

25. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

27. Bouncing Ball