Taylor and Maclaurin series

Let the order go to infinity: $f(x) = \sum_{k\ge0}\frac{f^{(k)}(a)}{k!}(x - a)^k$ whenever the remainder tends to 0. Functions equal to their Taylor series are called analytic; $e^x$, $\sin$, $\cos$ are, on all of $\R$.

A Taylor series packs a whole function into one sequence of numbers (its coefficients). That turns analysis into algebra: solving differential equations, evaluating integrals with no closed form, defining $e^A$ for a matrix $A$ or $e^{i\theta}$ for a complex number.

Knowing everything about $f$ at a single point — all its derivatives — is, for analytic functions, knowing $f$ everywhere within the radius of convergence. Not every smooth function is analytic: $e^{-1/x^2}$ (extended by 0) has all derivatives zero at 0, so its Taylor series is $0$, yet the function is not.

$\int_0^1 e^{-x^2}\,\dd x$ has no elementary antiderivative, but term-by-term integration of $e^{-x^2} = \sum (-1)^k x^{2k}/k!$ gives $\sum_k \frac{(-1)^k}{k!(2k+1)} = 1 - \frac13 + \frac1{10} - \frac1{42} + \dots \approx 0.746824$ — the core of the Gaussian error function erf.

Series let a computer evaluate transcendental functions, let a CAS manipulate them, and let physicists and engineers build perturbation methods. The matrix exponential $e^{At}$ — the solution of every linear ODE system and of every quantum gate $e^{-iHt}$ — is defined by its series.