04 // Interactive Formula Compendium // CALCULUS-ALGEBRAf(x) = Σ_{k=0..∞} [ f⁽ᵏ⁾(a) / k! ] · (x − a)ᵏ
Taylor & Maclaurin Series Expansion Formula — Polynomial Approximation
A Taylor series approximates a smooth function f(x) around an expansion point a via the power series P_n(x) = Σ_{k=0..n} [f⁽ᵏ⁾(a)/k!](x − a)ᵏ; when centered at a = 0, it is called a Maclaurin series.
Formula Reference
Main Formula
f(x) = Σ_{k=0..∞} [ f⁽ᵏ⁾(a) / k! ] · (x − a)ᵏ
Rearranged Forms
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + … (Maclaurin at a = 0)
sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + …
R_n(x) = [ f⁽ⁿ⁺¹⁾(ξ) / (n + 1)! ] · (x − a)ⁿ⁺¹ (Lagrange Remainder)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| P_n(x) | n-th Order Maclaurin Approximation of eˣ | dimensionless |
| x | Evaluation Input (x) | 1 |
| n | Polynomial Degree (n terms) | integer |
Inputs & Parameters
[1]
[integer]
Calculated Result (n-th Order Maclaurin Approximation of eˣ)
P_n(x) =2.716667dimensionless
Formula Notes & Background
- Require an n-th degree polynomial P_n(x) = c₀ + c₁(x−a) + c₂(x−a)² + … to match f(x) and its first n derivatives at x = a.
- Differentiating k times and evaluating at x = a eliminates all lower terms and leaves P_n⁽ᵏ⁾(a) = k! · c_k = f⁽ᵏ⁾(a).
- Solving for c_k yields c_k = f⁽ᵏ⁾(a) / k!, and applying the Mean Value Theorem n + 1 times produces the Lagrange error bound.
Step-by-Step Examples
#1: Approximate e¹ using a 5th-degree Maclaurin polynomial (k = 0 to 5).
x = 1n = 5f⁽ᵏ⁾(0) = 1 for all k
P₅(1) = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120
→ P₅(1) = 2.71667 (exact e = 2.71828, error = 0.00161)
#2: Approximate sin(0.5 rad) using the cubic Taylor polynomial x − x³/6.
x = 0.5 radn = 3
P₃(0.5) = 0.5 − (0.5)³ / 6 = 0.5 − 0.125 / 6
→ 0.47917 (exact sin(0.5) = 0.47943)
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Frequently Asked Questions
What is the difference between a Taylor series and a Maclaurin series?+
A Maclaurin series is simply the special case of a Taylor series where the expansion center is a = 0.
How do you bound the truncation error of an n-th degree Taylor polynomial?+
By Lagrange's Remainder Theorem, the absolute error is bounded by max|f⁽ⁿ⁺¹⁾(ξ)| × |x − a|ⁿ⁺¹ / (n + 1)! for ξ between a and x.
Why are first- and second-order Taylor expansions pervasive in physics and engineering?+
First-order expansions linearize nonlinear systems around an operating point (e.g. sin θ ≈ θ), while second-order expansions model harmonic potential wells near equilibrium.