Day 39 in MAT229

Today:

• Announcements:
  • Homework 11.4: grading #10, 20

  • Any more Homework 11.5 today?

• Section 11.6
  • Further Examples:
    • Consider the non-linear, first-order ODE
      • What happens (that seems different) when we attempt to solve the DE using power series?
      • How could we solve this analytically w/o power series?

    • #47, p. 586

• Section 11.7: Taylor Series
  • We've basically seen all this material before: we're now just summarizing:
    
    is the Taylor Series expansion of f(x) centered at x=c. The special case of c=0 is called the Maclaurin series. The factorials in the denominator sweep up all the powers that pop down during the n differentiations.

    The coefficients a_kare "designed" so that after k differentiations of the series, when we evaluate the series we pick off only the k^th derivative evaluated at x=c:

    

    

    We've seen that, because of the radius of convergence, the function may not be represented by a power series at some values of x (even as it's perfectly represented at other values of x).

    If we can bound all derivatives of a function on an interval about x=c, then the Taylor series represents the function there:

    

    The expansion for the binomial series is a classic: it's another of the great achievements of Sir Isaac Newton, one of the founders of calculus:

    

  • Examples:
    • #1, p. 596
    • #4, p. 596 (remember that there are several different ways to do these! You don't have to work from the definition....)