Expii

# 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 $$f(x) = T(x)$$.