Fórmula de Taylor com termo de erro de Lagrange, [;\mathcal{E}_n(t,\varepsilon);]

• Fórmula de Taylor com termo de erro de Lagrange, \(\mathcal{E}_n(t,\varepsilon), \varepsilon>0:\) \(f(t+\varepsilon) =\) \(f(t)+\varepsilon f\prime(t)+\frac{\varepsilon^2}{2} f\prime \prime(t)+...+\frac{\varepsilon^n}{n!} f^{(n)}(t)+\mathcal{E}_n(t,\varepsilon)=\) \(\sum_{i=0}^n \frac{ \varepsilon^i }{i!} f^{(i)} (t)+\mathcal{E}_n(t,\varepsilon)\sum_{i=0}^n \frac{ \varepsilon^i }{i!}= f^{(i)} (t)+\mathcal{E}_n(t,\varepsilon)\)
• \(\exists \xi \in [t,t+\varepsilon],\) tal que: \(|\mathcal{E}_n(t,\varepsilon)|\leq\frac{ \varepsilon^{n+1} }{ (n+1)! }| f^{(n+1)}(\xi) |\)
• Anote: \(\lim_{\varepsilon \rightarrow 0}\frac{ \mathcal{E}_n(t,\varepsilon) }{ \varepsilon^{n} }=0\)
• Escrevemos: \(o(\varepsilon^{n}):=\mathcal{E}_n(t,\varepsilon)\)
  \(f\prime(t)=\frac{ f(t+\varepsilon)-f(t)}{\varepsilon} +o(\varepsilon)\)