Formula di Taylor con Resto di Lagrange Enunciato Sia: f:(a,b)→R,x0∈(a,b), f∈Cn+1∈(a,b)f:(a,b)\to \mathbb{R}, x_0 \in (a,b),\ f \in C^{n+1} \in (a,b)f:(a,b)→R,x0∈(a,b), f∈Cn+1∈(a,b) ⇒∀x∈(a,b),x≠x0, ∃ϵ\\ \Rightarrow \forall x \in (a,b), x \neq x_0,\ \exists \epsilon ⇒∀x∈(a,b),x=x0, ∃ϵ compreso trax0,xx_0,xx0,x f(x)=∑k=0nfk(x0)k!⋅(x−x0)+f(n+ϵ)(ϵ)(n+1)!⋅(x−x0)n+1\\ f(x) = \sum_{k=0}^n \frac{f^k(x_0)}{k!}\cdot (x-x_0) + \frac{f^{(n+\epsilon )}(\epsilon)}{(n+1)!}\cdot (x-x_0)^{n+1} f(x)=∑k=0nk!fk(x0)⋅(x−x0)+(n+1)!f(n+ϵ)(ϵ)⋅(x−x0)n+1 Osservazioni ϵ\epsilonϵ dipende da x,x0,nx,x_0,nx,x0,n e con n=0⇒f(x)=f(x0)+f′(ϵ)(x−x0)n=0 \Rightarrow f(x) = f(x_0)+f'(\epsilon)(x-x_0)n=0⇒f(x)=f(x0)+f′(ϵ)(x−x0)