Definition and Taylor-Young’s formula

What is a limit series?

A limit series (LS) of a function f in the neighbourhood of a point a, of order n, is an approximate representation of f(x) in the form of a polynomial in (x-a), accompanied by a negligible remainder:

f(x) = a₀ + a₁(x − a) + a₂(x − a)² + … + a_n(x − a)^n + (x − a)^n * ε(x)

where ε(x) → 0 as x → a. The remainder is often written in Landau notation as o((x − a)^n).

Taylor–Young formula

If f is n-times differentiable at a, then it has a Taylor–Young series of order n given by:

f(x) = f(a) + f'(a)(x-a) + f''(a)/2! * (x–a)² + … + f^(n)(a)/n! * (x–a)^n + o((x–a)^n)

Example: The second-order Taylor series of f(x) = √(1+x) at 0.

f(0) = 1, f'(x) = 1/(2sqrt(1+x)), so f'(0) = 1/2. f''(x) = -1/4(1+x)^(-3/2), so f''(0) = -1/4.

Derivative: sqrt(1+x) = 1 + x/2 - x^2/8 + o(x^2)

Uniqueness of the DL

If a DL exists, it is unique: two polynomials that approximate the same function to within o((x-a)^n) are necessarily equal. This uniqueness means that any method (Taylor series, substitution, product rule, etc.) can be used to obtain it, regardless of the method chosen.

Common pitfall

Do not confuse ‘n-times differentiable’ with ‘an nth-order DL exists’: the existence of an nth-order DL does not imply that f is n-times differentiable, but the converse (Taylor–Young) holds as soon as f is of class C^n at a.