The Taylor series represents a function as an infinite sum of terms built from its derivatives evaluated at a point, with the general term given by the nth derivative at that point divided by n factorial, times the variable raised to the nth power. For a polynomial, this series terminates naturally because derivatives beyond the polynomial's degree vanish identically, so the Taylor series reproduces the original polynomial exactly rather than merely approximating it. This belongs to single-variable calculus, illustrating that a Taylor expansion is a generalized local approximation scheme whose fidelity to the original function follows directly from how many nonzero derivative terms exist.
The Taylor series represents a function as an infinite sum of terms built from its derivatives evaluated at a point, with the general term given by the nth derivative at that point divided by n facto…