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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
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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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
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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