Taylor's Theorem - Real Analysis | Lecture 28

In this lecture we state and prove Taylor's theorem. We begin by defining Taylor polynomials and then demonstrate that Taylor's theorem is entirely concerned with the error (or remainder) between a function and its Taylor polynomial. We emphasize that Taylor's result should be viewed as a generalization of the mean value theorem, and indeed its proof is quite similar as it relies upon Rolle's theorem. The lecture concludes with an illustrative example that shows that the exponential function can be approximated by a Taylor polynomial to any precision at any point in the real numbers. This course is taught by Jason Bramburger.