Taylor series rebuild a function from derivatives at one point
Brook Taylor introduced these expansions in 1715: an infinite sum whose coefficients are derivatives at a single center. Near that point many familiar functions match their series. Truncating to the first n plus one terms yields the nth Taylor polynomial, with Taylor's theorem bounding the leftover error.
In analysis the Taylor series of a function is an infinite sum built from that function's derivatives at one point. For many common maps the function equals its series near the expansion point. When the center is zero the series is also called a Maclaurin series, after Colin Maclaurin. The partial sum of the first n + 1 terms is a degree-n Taylor polynomial that generally approximates better as n grows. If the series converges, its sum is the limit of those polynomials, though a function may differ from its Taylor series even when the series converges.
For an infinitely differentiable real or complex-valued f at a point a, the series is sum over n of f to the n at a, over n factorial, times (x − a) to the n. The zeroth derivative is f itself, and both (x − a) to the 0 and 0! equal 1. At a = 0 the powers of x alone appear. Taylor series inherit power-series algebra: sums, differences, products, scalar multiples, and Cauchy products of series for products of functions. Compositions arise by substituting one convergent series into another when the substitution is valid. Termwise differentiation and integration keep the same radius of convergence, though endpoint behavior may change.
Those operations let one compute expansions such as arctangent from simpler series like the geometric series. Methods include applying the definition when a general derivative formula is known, or manipulating known expansions by substitution, multiplication, division, addition, subtraction, or termwise calculus. Differentiating 1/(1 − x) = 1 + x + x squared + ··· for |x| < 1 produces 1/(1 − x) squared as 1 + 2x + 3x squared + ···. Integrating the geometric series from 0 to x yields −log(1 − x) as x + x squared over 2 + x cubed over 3 + ···.
Composition examples include replacing t by −x squared in the geometric series to get 1/(1 + x squared) = 1 − x squared + x to the fourth − ···, then integrating termwise to recover the arctangent series with alternating odd powers. Higher Maclaurin polynomials for composed maps such as ln(cos x) can be built by nesting known expansions with remainder notation. The practical craft of Taylor series is therefore as much algebraic reuse of standard Maclaurin starters as brute-force differentiation at the center.
Source: Taylor series
