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 §10.2 Test Convergence

2. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §10.1 Infinite & Geometric Series  Any QUESTIONS About HomeWork §10.1 → HW-17 10.1

3. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 3 Bruce Mayer, PE Chabot College Mathematics §10.2 Learning Goals  Use Specialized Tests to Determine Convergence or Divergence of an Infinite Series Divergence test Integral test Ratio test

4. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 4 Bruce Mayer, PE Chabot College Mathematics ReCall Infinite SEQUENCE  Sequence Attributes a)An ORDERED LIST of Values b)Domain of the NATURAL Numbers Domain:1,2,3,4,…,n,n,… ↓↓↓↓↓ Range:a1,a1,a2,a2,a3,a3,a3,a3,…,an,an,…

5. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 5 Bruce Mayer, PE Chabot College Mathematics Sequential PATTERNS  Recognize a pattern for each sequence. Write a formula for the n th term

6. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 6 Bruce Mayer, PE Chabot College Mathematics Sequential CONvergence  Write the 1 st 5 terms for:  Graphically:  Note that as n becomes Larger the Sequence Elements CONVERGE on the Value 1 (one)

7. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 7 Bruce Mayer, PE Chabot College Mathematics Sequential DIvergence  Write the 1 st 5 terms for:  Graphically:  Note that as n becomes Larger the Sequence Elements do NOT approach a single Value; i.e., it DIVERGES

8. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 8 Bruce Mayer, PE Chabot College Mathematics ReCall Infinite Series  An Infinite Series Represents the Sum of the terms in a Sequence, e.g.;  An Infinite Series May: CONverge to a FINITE Value Diverge to +∞ or −∞  Next Develop TESTS for Series Divergence or Convergence

9. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 9 Bruce Mayer, PE Chabot College Mathematics Divergence Test  The FIRST requirement of convergence is that the TERMS must approach ZERO  Thus the: n th term test for divergence The Series  DIVERGES if either this Limit  –Does Not EXIST –Does Not EQUAL ZERO  Note that the n th term-test can prove that a series DIverges, but can not prove that a series CONverges. 

10. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 10 Bruce Mayer, PE Chabot College Mathematics Examples  Divergence Test  A Divergent Series  The Divergence Test does NOT However, Detect ALL Divergent Series. The HARMONIC Series Diverges, but it Fails the Test for Divergence

11. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 11 Bruce Mayer, PE Chabot College Mathematics Integral Test → Formal Statement  Let { a n } be a sequence of positive terms. Suppose that a n = f ( n ) where f is a continuous positive, decreasing function of x for all x  N. Then the series and the corresponding integral shown both CONverge OR both DIverge f(n)f(n) f(x)

12. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 12 Bruce Mayer, PE Chabot College Mathematics Integral Test → ReStated  Practically Speaking for the Integral Test if f ( x ) is a fcn such that f ( n ) = a n :  And finding Integral-Value is often MUCH EASIER that Finding Sum-Value Thus Check INTEGRAL First CONverges ifCONverges DIverges ifDIverges

13. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 13 Bruce Mayer, PE Chabot College Mathematics Integral Test GeoMetrically  The Area in each Rectangle Corresponds to Term in sequence  By Area Under Curve: CIRCUMscribed Rectangles INscribed Rectangles

14. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 14 Bruce Mayer, PE Chabot College Mathematics Integral Test GeoMetrically  Thus if the Area Under the Curve in Both cases is FINITE, then so is the SUM  The two integrals “Box-In” the value of the sum IF the Integrals are finite

15. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Integral Test  Consider Harmonic Series Σ ( 1 / k )  Which Confirms that:  And a VERY SIMILAR, but CONvergent Series: Verifying Convergence By Integral Test

16. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 16 Bruce Mayer, PE Chabot College Mathematics The p-Series Test

17. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 17 Bruce Mayer, PE Chabot College Mathematics p-Series Comparison  CONvergent  DIvergent Note Difference in AUC

18. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 18 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE % MTH-16 14Apr14 % MTH15_Quick_Plot_BlueGreenBkGnd_130911.m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = 1; xmax = 10; % The FUNCTION ************************************** x = linspace(xmin,xmax,50); yc = 1./x.^1.1; yd = 1./x.^0.9; % *************************************************** % the Plotting Range = 1.05*FcnRange ymin = min(yc); ymax = max(yc); % the Range Limits R = ymax - ymin; ymid = (ymax + ymin)/2; ypmin = ymid - 1.025*R/2; ypmax = ymid + 1.025*R/2 % % the Plot axes; set(gca,'FontSize',16); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green subplot(1,2,1); bar(x,yc, 'LineWidth', 1),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{20}n'), ylabel('\fontsize{20}y = f(x) = 1/x^1^.^1'),... title(['\fontsize{24}MTH16 Bruce Mayer, PE',]), subplot(1,2,2); bar(x,yd, 'LineWidth', 1),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{20}n'), ylabel('\fontsize{20}y = f(x) = 1/x^0^.^9'),... title(['\fontsize{24}MTH16 Bruce Mayer, PE',]

19. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 19 Bruce Mayer, PE Chabot College Mathematics Direct Comparison Test  Consider Two Series:  Where for all n : Note that b is the SMALL one  Then CONverges then so doesIf DIverges then so doesIf If the BIG one CONverges then so does the Small one If the SMALL one DIverges then so does the Big one

20. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Use Comparison Test  Determine Convergence Condition for this Series:  SOLUTION  The Divergence Test gives no useable information as  The Integral Test can NOT be used as NO AntiDerivative Exists for

21. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Use Comparison Test  The p-series test does not apply as the sequence is not of the form 1/ k p  However, note that for any k ≥ 0  Now by the p-series Test  And  Thus the series CONverges by Comparison to the Larger Quantity

22. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 22 Bruce Mayer, PE Chabot College Mathematics The Ratio Test  For the Series  Let  Then a)If L < 1, Then the Series CONverges b)If L > 1, Then the Series DIverges c)If L = 1, Then the Test is NOT Conclusive –When L = 1 the Convergence Condition of the series can NOT be discerned with the Ratio Test

23. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Ratio Test  Find the Convergence Behavior for →  Apply the Ratio Test  Thus the Limit exceeds 1 so the Series DIverges

24. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  UnKnown Situation

25. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  UnKnown Situation

26. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  UnKnown Situation  Perform the Analogous Integration

27. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  UnKnown Situation  So Finally  Thus the Integral converges to a finite number, and by the Integral Test the Series also Cverges.  Note by MATLAB & MuPAD

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

29. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 29 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE * 14Apr14 % clear; clc; clf; % N = 20 % the Number terms in the Sum N+1 for n = 0:N k = 2:n+2; terms = 4./(k.*(log(k)).^6) S(n+1) =sum(terms); end % Calc DIFFERENCE compare to pi/4 % % % The y = ZERO Lines zxh = [0 N]; zyh = [19.033 19.033]; axes; set(gca,'FontSize',12); plot((0:N),S,'k', 'LineWidth',1.5), grid,... xlabel('\fontsize{14}n'), ylabel('\fontsize{14}S_n'),... title(['\fontsize{16}MTH16 Convergent Sum',]) hold on plot(zxh,zyh, '-.k', 'LineWidth', 2) hold off

30. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 30 Bruce Mayer, PE Chabot College Mathematics MuPAD  float(Sn)  sum(4/(k*(ln(k))^6), k)  Sn := sum(4/(k*(ln(k))^6), k=2..202)

31. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 31 Bruce Mayer, PE Chabot College Mathematics WhiteBoard PPT Work  Problems From §10.2 P45 → 3-Point Bending Tests of 10 000 Randomly Selected Laminated Beams

32. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 32 Bruce Mayer, PE Chabot College Mathematics All Done for Today Series: Convergence Testing

33. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 33 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

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

35. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 35 Bruce Mayer, PE Chabot College Mathematics P45 → MINIMUM Break Strain  100 Laminated Beams in Random Jumble to be Tested for MINIMUM Breaking Force  Game Plan Break Beam-1 to find F BB = F 1 –Odds of Breaking Beam-1 = 100% Test Beam-2 until ONE of: a)BREAKING Beam-2; in which case F BB = F 2 b)F 2 = F BB withOUT Breaking –Odds of Breaking Beam-2 = 1 out of 2 = 50%

36. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 36 Bruce Mayer, PE Chabot College Mathematics P45 → MINIMUM Break Strain Test Beam-3 until ONE of: a)BREAKING Beam-3; in which case F BB = F 3 b)F 3 = F BB withOUT Breaking –Odds of Breaking Beam-3 = 1 out of 3 = 33% Test Beam-4 until ONE of: a)BREAKING Beam-4; in which case F BB = F 4 b)F 4 = F BB withOUT Breaking –Odds of Breaking Beam-4 = 1 out of 4 = 25%  Continuing with the plan up to beam-n

37. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 37 Bruce Mayer, PE Chabot College Mathematics P45 → MINIMUM Break Strain Test Beam-n until ONE of: a)BREAKING Beam-n; in which case F BB = F n b)F n = F BB withOUT Breaking –Odds of Breaking Beam-n = 1 out of n = 1/n  Then the TOTAL Number of Beams broken is Simply the SUM of the Individual Beam-Breaking Odds, i.e.;

38. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 38 Bruce Mayer, PE Chabot College Mathematics Broken Beam vs No. Beams

39. BMayer@ChabotCollege.edu MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 39 Bruce Mayer, PE Chabot College Mathematics %-Broken vs No. Beams