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

1. Chabot Mathematics §10.3 Series:Power & Taylor Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 10.2 Review § • Any QUESTIONS About • §10.2 Convergence Tests • Any QUESTIONS About HomeWork • §10.2 → HW-18

3. §10.3 Learning Goals • Find the radius and interval of convergence for a power series • Study term-by-term differentiation and integration of power series • Explore Taylor series representation of functions

4. Power Series • General Power Series: • A form of a GENERALIZED POLYNOMIAL • Power Series Convergence Behavior • Exclusively ONE of the following holds True • Converges ONLY for x= 0 (Trival Case) • Converges for ALL x • Has a Finite “Radius of Convergence”, R

5. Radius of Convergence • For the General Power Series • Unless a power series converges at any real number, a number R > 0 exists such that the series CONverges absolutely for each x such that | x | < R and DIverges for any other x • Thus the “Interval of Convergence”

6. Example  Radius of Conv. • Find R for the Series: • Radius of Convergence • Interval of Convergence • SOLUTION • Use the Ratio Test

7. Example  Radius of Conv. • Continue with Limit Evaluation: • Thus R = 4 • The Interval of Convergence • Thus This SeriesConverges

8. Functions as Power Series • Many Functions can be represented as Infinitely Long PolyNomials • Consider this Function and Domain • Recall one of The Geometric Series • Thus

9. Example  Fcn by Pwr Series • Write as a Power Series → • Also Find the Radius of Convergence • SOLUTION: • Start with the GeoMetric Series • First Cast the Fcn into the Form

10. Example  Fcn by Pwr Series • Using Algebraic Processes on the Fcn • Thus by the Geometric Series • Then the Function by Power Series

11. Example  Fcn by Pwr Series • Now find the Radius of Convergence by the Ratio Test

12. Example  Fcn by Pwr Series • Thus for Convergence • So the Interval of Convergence: • And also the Radius of Convergence

13. Pwr Series Derivatives & Integrals • Consider a Convergent Power Series • And an Associated Function • If f(x) is differentiable over −R<x<R, then

14. Pwr Series Derivatives & Integrals • If f(x) is Integrable over −R<x<R, then

15. Pwr Series Derivatives & Integrals • Thus the Derivative of a Power-Series Function • Thus the AntiDerivative of a Power-Series Function

16. Example  Find Fcn by Integ • Find a Power Series Equivalent for • SOLUTION: • First take: • Recognizefrom Before

17. Example  Find Fcn by Integ • Recover the Original Fcn by taking the AntiDerivative of the Just Determined Derivative

18. Example  Find Fcn by Integ • Then • To Find C use the original Function • Use f(0) = 0 in Power Series fcn • Then the Final Power Series Fcn

19. Taylor Series • Consider some general Function, f(x), that might be Represented by a Power Series • Thus need to find CoEfficients, an, such that the Power Series Converges to f(x) over some interval. Stated Mathematically Need an so that:

20. Taylor Series • If x = 0 and if f(0) is KNOWN then • a0 done, 1→∞ to go…. • Next Differentiate Term-by-Term • Now if the First Derivative (the Slope) is KNOWN when x = 0, then

21. Taylor Series • Again Differentiate Term-by-Term • Now if the 2ndDerivative (the Curvature) is KNOWN when x = 0, then

22. Taylor Series • Another Differentiation • Again if the 3rd Derivative is KNOWN at x = 0 • Recognizing the Pattern:

23. Taylor Series • Thus to Construct a Taylor (Power) Series about an interval “Centered” at x= 0 for the Function f(x) • Find the Values of ALL the Derivatives of f(x) when f(x) = 0 • Calculate the Values of the Taylor Series CoEfficients by • Finally Construct the Power Series from the CoEfficients

24. Example  Taylor Series for ln(e+x) • Calculate the Derivatives • Find the Values of the Derivatives at 0

25. Example  Taylor Series for ln(e+x) • Generally • Then the CoEfficients • The 1st four CoEfficients

26. Example  Taylor Series for ln(e+x) • Then the Taylor Series

27. Taylor Series at x ≠ 0 • The Taylor Series “Expansion” can Occur at “Center” Values other than 0 • Consider a function stated in a series centered at b, that is: • Now the the Radius of Convergence for the function is the SAME as before:

28. Taylor Series at x ≠ 0 • To find the CoEfficientsneed (x−b) = 0 which requires x = b, Then the CoEfficient Expression • The expansion about non-zero centers is useful for functions (or the derivatives) that are NOT DEFINED when x=0 • For Example ln(x) can NOT be expanded about zero, but it can be about, say, 2

29. Example  Expand x½ about 4 • Expand about b = 4: • The 1st four Taylor CoEfficients

30. Example  Expand x½ about 4 • SOLUTION: • Use the CoEfficients to Construct the Taylor Series centered at b = 4

31. Example  Expand x½ about 4 • Use the Taylor Series centered at b = 4 to Find the Square Root of 3

32. WhiteBoard PPT Work • Problems From §10.3 • P39 → expand aboutb = 1 the Function

33. All Done for Today BrookTaylor (1685-1731)

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

36. P10.3-39 Taylor Series • Da1 := diff(ln(x)/x, x) • Db2 := diff(Da1, x) • Dc3 := diff(Db2, x) • Dd4 := diff(Dc3, x)

37. P10.3-39 Taylor Series • ln(x)/x, x • f0 := taylor(ln(x)/x, x = 1, 0) • f1 := taylor(ln(x)/x, x = 1, 1) • f2 := taylor(ln(x)/x, x = 1, 2)

38. P10.3-39 Taylor Series • f3 := taylor(ln(x)/x, x = 1, 3) • f4 := taylor(ln(x)/x, x = 1, 4) • d6 := diff(ln(x)/x, x $ 5)

39. P10.3-39 Taylor Series • plot(f0, f1, f2, f3, f4, f5, x =0.5..3, GridVisible = TRUE,LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24],TicksLabelFont=["sans-serif", 16])