Where Does a Function Equal Its Taylor Series? - Expii

To analyze whether a function \(f(x)\) equals its Taylor series \(T(x)\) centered at \(x=a\) on an interval \((a-R,a+R)\), we can often use the Lagrange error bound to study the remainder \(R_n(x) = f(x) - T_n(x)\) for each degree \(n\) as \(n\to\infty\). If \(\lim_{n\to\infty} R_n(x) = 0\) for a given value of \(x\), then \(f(x) = T(x)\).