Taylor's theorem with remainder

Taylor expansion with an explicit remainder term for a smooth real function.

Taylor’s theorem with remainder

Taylor’s theorem with remainder: Let $I\subseteq\mathbb R$ be an interval , let $a\in I$ , and let $f:I\to\mathbb R$ have continuous derivatives up to order $n+1$ on $I$ . For each $x\in I$ there exists a point $\xi$ between $a$ and $x$ such that

f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}.

The polynomial part is the Taylor polynomial of degree $n$ at $a$ , and the final term is the Lagrange-form remainder, which provides effective error bounds when $f^{(n+1)}$ is bounded.