Calculus Small Changes, Big Ideas

Section 10.2 Error in Taylor series

• Taylor’s theorem: For any other \(x_{0}\in I\) there exists some \(\alpha_{n}\) between \(x_{0}\) and \(c\) (which depends on \(x_{0}\) and \(c\) and \(n\)) such that

  \begin{equation*} f(x_{0})=\qty(\sum_{k=0}^{n-1}\frac{f\psup k(c)}{k!}(x_0-c)^k)+\frac{f\psup n(\alpha_{n})}{n!}(x_{0}-c)^{n}\text{.} \end{equation*}

  That is, \(R_{n-1}(x_{0})=\dfrac{f\psup n(\alpha_{n})}{n!}(x_{0}-c)^{n}\text{,}\) for some \(\alpha_{n}\) between \(x_{0}\) and \(c\text{.}\)

  🔗
  🔗
• Integral form of remainder function:

  \begin{equation*} R_{n}(x)=\frac{1}{n!}\int_{c}^{x}(x-t)^{n}f\psup{n+1}(t)\dd{t} \end{equation*}

  🔗
  🔗
• Lagrange error bound: If \(|f\psup{n+1}(x)|\le M\) for \(|x-a|\le d\text{,}\) then

  \begin{equation*} |R_{n}(x)|\le\frac{M}{(n+1)!}|x-a|^{n+1} \end{equation*}

  for \(|x-a|\le d\text{.}\)

  🔗
  🔗