The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 3. APPROXIMATE SOLUTIONS 47
Figure 3.1: Solution curves for example 3.1. The curves correspond to h = 1 (red);
h = 1/2 (green); h = 1/4 (blue); h = 1/8 (purple); and the exact solution (orange).
Taylor Series
Theorem 3.1. Taylor’s Theorem with Remainder. Let f : J ↦→ R be C n+1 on
J ⊂ R, and suppose that a ∈ interior(J). Then for any x ∈ J, there exists some
number c ∈ J, where c is between a and x, inclusive, such that
f(x) = P n (x) + R n (x) (3.34)
where the The Taylor Polynomial of order n for f about x =a is
and
P n (x) = f(a) + (x − a)f ′ (a) + (x − a)2 f ′′ (a)
2!
R n (x) = (x − a)n+1 f (n+1) (c)
(n + 1)!
Theorem 3.2. If, in addition, f ∈ C ∞ then
+ · · · + (x − a)n f (n) (a)
) (3.35)
n!
(3.36)
∞∑ f k (a)(x − a) k
f(x) =
k!
k=0
(3.37)
which is called the Taylor Series of f about x =a.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge