Taylor Series of Analytic Functions: Radius and Examples

A Taylor series rewrites an analytic function as an infinite sum of powers of (z − a), built from the function's derivatives at the point a: f(z) = f(a) + f′(a)(z − a) + f″(a)(z − a)²/2! + f‴(a)(z − a)³/3! + …. Truncating the sum after a few terms gives a polynomial approximation, while the full infinite series reproduces the function exactly wherever it converges.

This matters because an analytic function is completely determined by its derivatives at a single point: knowing f(a), f′(a), f″(a), … tells you the values of f throughout a whole disc around a. Physicists use this constantly — the small-angle approximation sin x ≈ x is just the first term of a Taylor series — and numerical software evaluates functions like eˣ and cos x using truncated Taylor polynomials.

Key Concepts

• Analytic at a: f is differentiable at every point of some neighbourhood of a.
• Taylor coefficients: the coefficient of (z − a)ⁿ is aₙ = f⁽ⁿ⁾(a)/n!, where f⁽ⁿ⁾ is the nth derivative.
• Radius of convergence R: the series converges for |z − a| < R and diverges for |z − a| > R. For analytic f, R equals the distance from a to the nearest singularity of f.
• Disc of convergence: the open disc |z − a| < R inside which the series is guaranteed to converge to f(z).
• Maclaurin series: a Taylor series centred at a = 0.

Formulas

• Taylor expansion: f(z) = Σ (n = 0 to ∞) f⁽ⁿ⁾(a)/n! · (z − a)ⁿ, valid for |z − a| < R.
• Radius (Cauchy–Hadamard): 1/R = lim sup |aₙ|¹/ⁿ as n → ∞, where aₙ are the Taylor coefficients.
• Standard series: eᶻ = Σ zⁿ/n!, cos z = Σ (−1)ⁿz²ⁿ/(2n)!, sin z = Σ (−1)ⁿz²ⁿ⁺¹/(2n+1)!, all with R = ∞; 1/(1 − z) = Σ zⁿ with R = 1.
• Assumptions and limits: f must be analytic throughout the disc; convergence on the boundary circle |z − a| = R must be checked separately, since it varies case by case.

Worked Example

Find the Maclaurin series of f(z) = eᶻ and its radius of convergence, then approximate e using 6 terms.

1. Every derivative of eᶻ is eᶻ, so f⁽ⁿ⁾(0) = e⁰ = 1 for all n.
2. Coefficients: aₙ = 1/n!, giving eᶻ = 1 + z + z²/2! + z³/3! + z⁴/4! + ….
3. Ratio test: |aₙ₊₁(z)ⁿ⁺¹| / |aₙzⁿ| = |z|/(n + 1) → 0 as n → ∞ for every fixed z, so the series converges for all z and R = ∞.
4. At z = 1 with six terms: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.71666….
5. The true value e = 2.71828…, so six terms already give an error under 0.002.

Common Misconceptions

• "Taylor series always converge everywhere." False. The series for 1/(1 + z²) centred at 0 has R = 1 because of singularities at z = ±i, even though the function looks perfectly smooth on the real line.
• "Any infinitely differentiable function equals its Taylor series." False for real functions: the classic counterexample is flat at 0 with all derivatives zero there, so its Taylor series is 0 while the function is positive nearby. (For complex analytic functions, equality on the disc always holds.)
• "Convergence inside the disc extends to the boundary." Not automatically. The series Σ zⁿ/n² converges at z = 1 but Σ zⁿ diverges there; each boundary point needs its own test.

Real-World Applications

• Physics approximations: pendulum motion, relativity corrections, and quantum perturbation theory all start by truncating a Taylor series.
• Numerical computing: calculators and math libraries evaluate transcendental functions with Taylor-like polynomial approximations.
• Differential equations: the Frobenius method solves ODEs by assuming a power-series solution and matching coefficients.

A Taylor series turns local derivative data at one point into global knowledge of the function across its disc of convergence. The radius is set by the nearest singularity, the coefficients come from f⁽ⁿ⁾(a)/n!, and truncated series give the polynomial approximations behind much of applied mathematics.