Taylor and Maclaurin Series
Reference for Taylor and Maclaurin series expansions.
Covers common series for e^x, sin(x), cos(x), ln(1+x), and the geometric series with convergence radii.
Taylor Series Formula
f(x) = Σ (n=0 to ∞) f⁽ⁿ⁾(a) / n! · (x - a)ⁿ
This expands any infinitely differentiable function f(x) around a point a.
Variables
| Symbol | Meaning |
|---|---|
| f⁽ⁿ⁾(a) | The n-th derivative of f evaluated at point a |
| n! | n factorial (n × (n-1) × ... × 1) |
| a | The center point of expansion |
| (x - a)ⁿ | The n-th power of the distance from center |
Maclaurin Series (Taylor series at a = 0)
f(x) = Σ (n=0 to ∞) f⁽ⁿ⁾(0) / n! · xⁿ
Common Maclaurin Series
| Function | Series | Converges for |
|---|---|---|
| e^x | 1 + x + x²/2! + x³/3! + ... | All x |
| sin(x) | x - x³/3! + x⁵/5! - x⁷/7! + ... | All x |
| cos(x) | 1 - x²/2! + x⁴/4! - x⁶/6! + ... | All x |
| ln(1+x) | x - x²/2 + x³/3 - x⁴/4 + ... | -1 < x ≤ 1 |
| 1/(1-x) | 1 + x + x² + x³ + ... | |x| < 1 |
| (1+x)^n | 1 + nx + n(n-1)x²/2! + ... | |x| < 1 |
| arctan(x) | x - x³/3 + x⁵/5 - x⁷/7 + ... | |x| ≤ 1 |
Example 1 — Expanding e^x
Write the first 5 terms of e^x
All derivatives of e^x equal e^x. At x=0: f⁽ⁿ⁾(0) = 1 for all n.
e^x = 1 + x + x²/2 + x³/6 + x⁴/24 + ...
e^1 ≈ 1 + 1 + 0.5 + 0.1667 + 0.0417 = 2.7083 (actual: 2.7183)
Example 2 — Approximating sin(0.1)
Use 3 terms of the Maclaurin series for sin(x)
sin(x) ≈ x - x³/6 + x⁵/120
sin(0.1) ≈ 0.1 - 0.001/6 + 0.00001/120
≈ 0.1 - 0.000167 + 0.000000083 = 0.099833
Actual value: 0.099833 (extremely accurate with just 3 terms!)
Key Notes
- Formula: f(x) = Σ f⁽ⁿ⁾(a)/n! × (x−a)ⁿ: Expands a function as an infinite polynomial around the point x = a. When a = 0, it is called a Maclaurin series. The series is exact (not an approximation) wherever it converges.
- Key Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + … (converges everywhere); sinx = x − x³/3! + x⁵/5! − … ; cosx = 1 − x²/2! + x⁴/4! − … ; 1/(1−x) = 1 + x + x² + … (converges for |x| < 1).
- Radius of convergence: Each Taylor series has a radius of convergence R — the series only equals the function within |x − a| < R. Outside this radius, the partial sums diverge. For eˣ, R = ∞; for ln(1+x), R = 1.
- Taylor polynomial error — Lagrange remainder: The error from truncating after n terms is |Rₙ(x)| ≤ M|x−a|ⁿ⁺¹/(n+1)! where M bounds the (n+1)th derivative. This bounds how accurate a finite polynomial approximation is.
- Applications: Taylor series are used in numerical methods (approximating sin, cos, eˣ in calculators and computers), physics (small-angle approximation sinθ ≈ θ), deriving limits of the 0/0 indeterminate form, and solving differential equations via power series methods.