Home/Calculus II/Power series and Taylor series/Taylor remainderSection 15.4Taylor remainderLagrange form:Copy LaTeXRN(x)=f(N+1)(ξ)(N+1)!(x−a)N+1R_N(x)=\frac{f^{(N+1)}(\xi)}{(N+1)!}(x-a)^{N+1}RN(x)=(N+1)!f(N+1)(ξ)(x−a)N+1View detail →for some ξ\xiξ between aaa and xxx. If ∣f(N+1)(t)∣≤M|f^{(N+1)}(t)|\le M∣f(N+1)(t)∣≤M, thenFormulaCopy LaTeX∣RN(x)∣≤M∣x−a∣N+1(N+1)!.|R_N(x)|\le\frac{M|x-a|^{N+1}}{(N+1)!}.∣RN(x)∣≤(N+1)!M∣x−a∣N+1.View detail →
Lagrange form:Copy LaTeXRN(x)=f(N+1)(ξ)(N+1)!(x−a)N+1R_N(x)=\frac{f^{(N+1)}(\xi)}{(N+1)!}(x-a)^{N+1}RN(x)=(N+1)!f(N+1)(ξ)(x−a)N+1View detail →
FormulaCopy LaTeX∣RN(x)∣≤M∣x−a∣N+1(N+1)!.|R_N(x)|\le\frac{M|x-a|^{N+1}}{(N+1)!}.∣RN(x)∣≤(N+1)!M∣x−a∣N+1.View detail →