Theorem 1.6 (Taylor's series for a function of one variable) If f(x) is continuous and possesses continuous derivatives of order n in an interval that incules I = a, then in that interval (x) = F(a) + (x - a)f (a) + (x - a) (x - a)"-1 (n - 1)! -f"(a) + ... + 21 -rn-)(a) + R,(x). where R(x), the remainder term. can be expressed in the form R.(x) = ((). (x- a)" a < { < x. n!
Theorem 1.6 (Taylor's series for a function of one variable) If f(x) is continuous and possesses continuous derivatives of order n in an interval that incules I = a, then in that interval (x) = F(a) + (x - a)f (a) + (x - a) (x - a)"-1 (n - 1)! -f"(a) + ... + 21 -rn-)(a) + R,(x). where R(x), the remainder term. can be expressed in the form R.(x) = ((). (x- a)" a < { < x. n!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Prove the given theorem and provide (1) examples.
Theorem 1.6 (Taytor's sertes for a function of one variable) If f(x) is continuous and possesses continuous
derivatives of order n in an interval that incules 1 = a, then in that interval
(x - a)"-
(n - 1)!
(x) = F(a) + (x - a)f"(a) + -a)
tCa) +..+
21
-rn-)(a) + R,(x).
where R,(x). the remainder term. can be expressed in the form
(x- a)" f(m)(E).
R.(x) =
a<{<x.
nt
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