Taylor and Maclaurin series
A machine that can only add and multiply still reports e^{0.7} and \sin 0.3. How can a function be rebuilt from its derivatives at a single point, and how far can we trust the result?
Where you are 5 inside
What this area is about
A tangent line matches a function’s value and slope at one point. A polynomial can also match the bending, and every higher derivative, and the coefficients that do it are forced: \displaystyle \frac{f^{(k)}(a)}{k!}. Keep every term and the result is a power series. It equals the function at exactly those x where the remainder f(x) - P_n(x) tends to 0 as n grows.
How these fit together
Definition. Taylor polynomials builds the degree n polynomial that matches f and its first n derivatives at the center a.
Results. The common Maclaurin series writes down the series for e^x, \sin x, \cos x and a few others, and builds new ones by substitution, multiplication and term by term calculus. The binomial series expands (1 + x)^k for any real k, valid for \lvert x \rvert < 1. The Lagrange error bound says that if \lvert f^{(n+1)} \rvert \le M between a and x, then \displaystyle \lvert f(x) - P_n(x) \rvert \le \frac{M\lvert x - a \rvert^{n+1}}{(n+1)!}.
Procedure. Using series to evaluate limits and integrals integrates a series term by term inside its interval of convergence to estimate integrals with no usable antiderivative, and settles hard limits.
A path through it
Taylor polynomials are the natural start, then the common Maclaurin series. Both the error bound, which proves those series equal their functions, and using series for limits and integrals come after them. The binomial series fits anywhere after Taylor polynomials.
Later, beyond this course, come Taylor’s theorem with remainder, power series solutions and moment generating functions.