Let $f$ have derivatives up to order $n$ at $x=c$. The $n$-th degree Taylor polynomial of $f$ centered at $c$ is the partial sum

The remainder (or error) after $n$ terms is defined as

The remainder measures how well $T_n(x)$ approximates $f(x)$: the Taylor series of $f$ converges to $f(x)$ at a point $x$ precisely when $R_n(x) \to 0$ as $n \to \infty$.