A Taylor series represents a smooth function as an infinite sum of simple polynomial terms, built entirely from the function's value and derivatives at a single chosen point. Named after Brook Taylor (1715), this technique underlies how calculators compute sine and cosine, how physicists build workable approximations of complicated equations, and how computers efficiently evaluate functions that have no simpler closed-form expression.
Curves like sin(x) or eˣ can look complicated, but a remarkable fact makes them far more manageable: near any single point, such a curve can be approximated arbitrarily well using just polynomials — the simplest, most computationally friendly functions in all of mathematics, built from nothing but addition, multiplication, and powers of x. A Taylor series is the specific infinite polynomial recipe for doing this, built entirely from the original function's derivatives (see the Derivative entry) at one chosen point.
There's no simple algebraic formula for sin(37°) — but calculators and computers don't need one. Internally, they compute sin(x) using a Taylor series approximation (or a closely related, highly optimised numerical technique descended from the same underlying idea), adding up just enough polynomial terms to reach the display's required precision, all within a fraction of a millisecond.
The Taylor series for eˣ is remarkably simple: 1 + x + x²/2! + x³/3! + x⁴/4! + ... (where n! denotes n factorial). Setting x=1 gives an infinite sum that converges to Euler's number e ≈ 2.71828... — connecting one of mathematics' most important constants directly to an elegant, easily stated infinite sum.
For a function f that is infinitely differentiable at a point a, its Taylor series centred at a is:
Each successive term uses one higher derivative, evaluated at the chosen centre point a, divided by the appropriate factorial.
A Taylor series centred specifically at a=0 is called a Maclaurin series (named after Colin Maclaurin, who popularised this special case in the 1740s, though Taylor's original 1715 work already covered it as well). Several of the most famous and useful series in mathematics are Maclaurin series:
A Taylor series does not necessarily converge to the correct function value for every x — it typically only converges within a certain radius of convergence around the centre point a (see the Series & Convergence entry). For eˣ, sin(x), and cos(x), this radius happens to be infinite (the series works for every real x); but for 1/(1−x), the radius is exactly 1, and the series fails to converge at all for |x|≥1, despite the original function 1/(1−x) being perfectly well-defined for many such values.
In practice, computers and calculators use a Taylor polynomial — the Taylor series truncated after finitely many terms — rather than a genuine infinite sum. Taylor's theorem provides a precise, rigorous bound on the error introduced by this truncation, allowing engineers and programmers to guarantee a desired level of accuracy while using only as many terms as are actually necessary for that purpose.
A function is called analytic at a point if its Taylor series there actually converges to the function's true value in some neighbourhood of that point — not every infinitely differentiable function has this property, surprisingly. Most familiar functions from calculus (polynomials, exponentials, trigonometric functions, and their combinations) are analytic everywhere they're defined, but genuine exceptions exist.
The function f(x)=e^(−1/x²) for x≠0, with f(0)=0, is infinitely differentiable everywhere, including at x=0 — yet every single derivative of f at x=0 evaluates to exactly zero. This means its Taylor series centred at 0 is simply the zero function (0+0x+0x²+...) — which certainly does NOT equal the original function f(x) for any x≠0. This striking, historically important counterexample shows that being infinitely differentiable is not sufficient, by itself, to guarantee a function equals its own Taylor series.
Joseph-Louis Lagrange provided a precise formula (the Lagrange remainder) expressing exactly the error between a function and its truncated Taylor polynomial approximation, in terms of a higher derivative evaluated at some unspecified point within the relevant interval. This remainder formula is the rigorous mathematical tool that converts an informal "Taylor series should be a good approximation" intuition into a precise, provable numerical error bound — essential for any serious scientific or engineering application relying on truncated series approximations.
Brook Taylor published the general series bearing his name in 1715, though special cases had been discovered earlier and independently by James Gregory (1670s) and Isaac Newton, and the underlying idea has some further antecedents in the work of Indian mathematician Mādhava of Sangamagrama (c. 1400 CE — see that entry), whose series for sine, cosine, and arctangent are, in modern terms, specific instances of what would later be called Taylor/Maclaurin series, discovered roughly three centuries before their more widely known European namesakes. The general theory, with the full rigorous remainder term, was substantially developed later by Lagrange (1797) and given fully modern rigorous foundations in the 19th century alongside the broader rigorisation of calculus (see the Limit and Continuity entries).
As explored in the Complex Number entry, a complex-differentiable (holomorphic) function is automatically analytic — its Taylor series always converges to the function itself within its radius of convergence, a dramatically stronger and cleaner guarantee than the real-number case (where, as shown above, infinitely differentiable functions can still fail to equal their own Taylor series). This striking difference between real and complex analysis is one of the most celebrated and initially surprising features distinguishing the two fields.