What is Taylor Series? The Complete 2025 Guide
The Taylor Series is one of the most powerful concepts in mathematics, transforming complex functions into infinite polynomials that even computers can understand. First introduced by Brook Taylor in 1715, these series have become essential tools in physics, engineering, computer science, and economics. In this comprehensive 3500-word guide, we'll demystify Taylor Series completely—from the fundamental formula to real-world applications and common misconceptions.
📑 Table of Contents
- 1. What Exactly is a Taylor Series?
- 2. The Taylor Series Formula Demystified
- 3. Maclaurin Series: The Special Case
- 4. Common Taylor Series You Must Know
- 5. Convergence & Radius of Convergence
- 6. The Remainder Term (Error Analysis)
- 7. Real-World Applications
- 8. Step-by-Step Calculation Examples
- 9. Using Taylor Series Calculators
- 10. Historical Background
- 11. Frequently Asked Questions
1. What Exactly is a Taylor Series?
A Taylor Series is an infinite sum of terms that approximates a function near a specific point, using the function's derivatives at that point. Think of it as a mathematical "recipe" that tells you how to build any smooth function from simpler polynomial pieces.
If you know everything about a function at a single point—its value, slope, curvature, and all higher-order behaviors—you can reconstruct what the function looks like in a neighborhood around that point. This is the fundamental insight behind Taylor Series.
Simple Analogy: Imagine you're trying to describe a curvy mountain road to someone. You could:
- Zero-order approximation: "The road is at height 1000m at mile marker 10."
- First-order (linear): "It's at 1000m and rising at 5% grade at marker 10."
- Second-order (quadratic): "It's at 1000m, rising at 5%, but the slope is decreasing at 0.1% per mile."
Each additional piece of information (derivative) gives you a better approximation of what happens as you move away from your reference point.
2. The Taylor Series Formula Demystified
The general Taylor Series expansion of a function \( f(x) \) about point \( a \) is:
Or more compactly using summation notation:
The function value at the center point. This is your starting reference height.
The slope term tells you how the function changes linearly as you move away from \( a \).
The curvature term accounts for how the slope itself changes (concavity).
The \( n! \) in the denominator isn't arbitrary—it comes from repeatedly integrating the constant function 1. Each time you integrate \( (x-a)^n \), you get \( \frac{(x-a)^{n+1}}{n+1} \), and after \( n \) integrations starting from the nth derivative, you end up with \( n! \) in the denominator.
3. Maclaurin Series: The Special Case
When the expansion point \( a = 0 \), the Taylor Series simplifies to a Maclaurin Series, named after Scottish mathematician Colin Maclaurin (though he acknowledged Taylor's priority).
Important Note: Every Maclaurin series is a Taylor series (with \( a=0 \)), but not every Taylor series is a Maclaurin series. Maclaurin series are particularly useful for functions that behave nicely near zero.
| Function | Maclaurin Series | First 4 Terms |
|---|---|---|
| \( e^x \) | \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \) | \( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \) |
| \( \sin(x) \) | \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \) | \( x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \cdots \) |
| \( \cos(x) \) | \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \) | \( 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \) |
4. Common Taylor Series You Must Know
These expansions appear so frequently that every calculus student should memorize them. They form the building blocks for more complex series.
\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \quad (\text{all } x) \]
Sine Function:
\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \quad (\text{all } x) \]
Cosine Function:
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \quad (\text{all } x) \]
Natural Logarithm:
\[ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (-1 < x \leq 1) \]
Geometric Series:
\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots \quad (|x| < 1) \]
Notice the patterns: eˣ has all positive terms with factorials in denominators. sin x has only odd powers with alternating signs. cos x has only even powers with alternating signs. ln(1+x) has all powers without factorials, alternating signs.
5. Convergence & Radius of Convergence
Taylor Series don't always work everywhere. The radius of convergence tells you how far from the center point the series actually equals the original function.
Radius of Convergence (R): The distance from the center point within which the series converges.
Visual representation of convergence radius
| Function | Center (a) | Radius of Convergence | Converges For |
|---|---|---|---|
| \( e^x \) | 0 | ∞ | All real x |
| \( \sin x, \cos x \) | 0 | ∞ | All real x |
| \( \ln(1+x) \) | 0 | 1 | \( -1 < x \leq 1 \) |
| \( \frac{1}{1-x} \) | 0 | 1 | \( |x| < 1 \) |
| \( \arctan x \) | 0 | 1 | \( |x| \leq 1 \) |
Even if a Taylor series converges, it might not converge to the original function! Functions that equal their Taylor series everywhere within the radius of convergence are called analytic functions. Most "nice" functions (polynomials, exponentials, trigonometric functions) are analytic, but there exist functions with Taylor series that converge but not to the function (except at the center point).
6. The Remainder Term (Error Analysis)
In practice, we use finite Taylor polynomials, not infinite series. The remainder term tells us how much error we make by truncating the series.
Where \( R_n(x) \) is the remainder after \( n \) terms. The Lagrange form of the remainder is:
for some \( c \) between \( a \) and \( x \).
sin(0.1) ≈ 0.1 - (0.1)³/6 + (0.1)⁵/120
= 0.1 - 0.0001666667 + 0.0000008333
= 0.0998341666
The actual value: sin(0.1) ≈ 0.0998334166
Error: |0.0998341666 - 0.0998334166| ≈ 0.00000075
Lagrange remainder bound: |R₃| ≤ (0.1)⁷/5040 ≈ 1.98×10⁻¹² (much smaller than actual error because our bound is conservative)
7. Real-World Applications
Taylor Series aren't just mathematical curiosities—they're essential tools across science and engineering.
Small angle approximations: sin θ ≈ θ, cos θ ≈ 1 - θ²/2 for pendulum motion, optics.
Relativistic corrections: \( \frac{1}{\sqrt{1-v²/c²}} ≈ 1 + \frac{v²}{2c²} \) for v ≪ c.
Function approximation: Calculators use Taylor series for sin, cos, exp, log functions.
Numerical methods: Finite difference approximations for derivatives in differential equations.
Utility functions: Approximating complex utility curves for risk analysis.
Option pricing: Taylor expansions in Black-Scholes and other financial models.
\[ T = 2\pi \sqrt{\frac{L}{g}} \left[ 1 + \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right] \]
For small angles (θ₀ < 0.2 rad ≈ 11°), we use the first term only:
\[ T \approx 2\pi \sqrt{\frac{L}{g}} \]
This is the small-angle approximation from the Taylor series of the exact elliptic integral solution.
8. Step-by-Step Calculation Examples
Let's walk through building a Taylor series from scratch for a moderately complex function.
eᵘ = 1 + u + u²/2! + u³/3! + u⁴/4! + ...
Step 2: Substitute u = -x²
e⁻ˣ² = 1 + (-x²) + (-x²)²/2! + (-x²)³/3! + (-x²)⁴/4! + ...
Step 3: Simplify each term
= 1 - x² + x⁴/2 - x⁶/6 + x⁸/24 - ...
Step 4: General term pattern
e⁻ˣ² = Σ (-1)ⁿ x²ⁿ / n! for n = 0 to ∞
Step 5: First 5 terms
e⁻ˣ² ≈ 1 - x² + x⁴/2 - x⁶/6 + x⁸/24
f(x) = ln(x) → f(1) = 0
f'(x) = 1/x → f'(1) = 1
f''(x) = -1/x² → f''(1) = -1
f'''(x) = 2/x³ → f'''(1) = 2
f⁽⁴⁾(x) = -6/x⁴ → f⁽⁴⁾(1) = -6
Step 2: Apply Taylor formula with a = 1
ln(x) = 0 + 1·(x-1) + (-1)(x-1)²/2! + 2(x-1)³/3! + (-6)(x-1)⁴/4! + ...
Step 3: Simplify
ln(x) = (x-1) - (x-1)²/2 + (x-1)³/3 - (x-1)⁴/4 + ...
Step 4: Pattern emerges
ln(x) = Σ (-1)ⁿ⁺¹ (x-1)ⁿ / n for n = 1 to ∞
9. Using Taylor Series Calculators
Modern tools like our Taylor Series Calculator automate the tedious derivative calculations and provide visualization.
1. Use parentheses generously: sin(x²) vs (sin x)² are completely different.
2. Start with low degree: Try n=3 or 5 first, then increase for more accuracy.
3. Check convergence: If your x is far from a, you'll need many terms for good approximation.
4. Use the step-by-step mode to learn, not just get answers.
Our Calculator: Shows all derivative steps, remainder term, and convergence analysis.
Perfect for: Students verifying homework, understanding the process.
Mathematica/WolframAlpha: Handles extremely complex functions, symbolic manipulation.
Perfect for: Researchers, advanced physics/engineering work.
Python SymPy: Library for symbolic mathematics, can generate Taylor series programmatically.
Perfect for: Developers, computational science.
10. Historical Background
The story of Taylor Series is a fascinating journey through mathematical history.
1712: Brook Taylor writes to John Machin about his "new method for incrementing quantities."
1715: Taylor publishes "Methodus Incrementorum Directa et Inversa" containing the general series formula.
1742: Colin Maclaurin publishes "Treatise of Fluxions" emphasizing the special case a=0, making it popular.
1755: Leonhard Euler applies Taylor series extensively in calculus, making them fundamental tools.
19th Century: Augustin-Louis Cauchy rigorously studies convergence, remainder terms, and analytic function theory.
Despite the name "Maclaurin series," Colin Maclaurin himself credited Brook Taylor in his 1742 treatise. The special case with a=0 appears in earlier works by Gregory, Leibniz, and Newton, but Maclaurin's systematic treatment made it widely known.
11. Frequently Asked Questions
Q: What's the difference between Taylor series and Taylor polynomial?
A: A Taylor series is the infinite sum. A Taylor polynomial is a finite truncation (first n terms) of that series. We use polynomials for actual computation since we can't sum infinitely many terms.
Q: Do all functions have Taylor series?
A: No. A function must be infinitely differentiable at the expansion point to have a Taylor series. Even then, the series might not converge, or might converge to something other than the function (except at the center point). Functions that equal their Taylor series are called analytic functions.
Q: How many terms do I need for a good approximation?
A: It depends on the function and how far x is from a. For eˣ or sin x near 0, 5-10 terms give excellent accuracy. For functions like ln(1+x) near the boundary of convergence, you might need hundreds of terms. Use the remainder formula to estimate error.
Q: Can Taylor series approximate discontinuous functions?
A: Not directly. Taylor series approximate smooth functions. For discontinuous functions or functions with discontinuous derivatives, you might use Fourier series instead, or piecewise Taylor expansions on each smooth interval.
Q: Why are Taylor series centered at 0 so common?
A: Three reasons: 1) Simplicity—the formulas are cleaner. 2) Many important functions have symmetry or special properties at 0. 3) In practice, you can often shift your coordinate system to make the point of interest become 0.
🚀 Try Our Taylor Series Calculator!
Don't just read about Taylor series—experiment with them! Our free calculator shows every derivative step, visualizes the approximation, and explains convergence.
Launch Taylor Series Calculator →References & Further Reading
- Taylor, B. (1715). Methodus Incrementorum Directa et Inversa. London.
- Maclaurin, C. (1742). Treatise of Fluxions. Edinburgh.
- Apostol, T. M. (1967). Calculus, Volume 1. John Wiley & Sons.
- Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage Learning.
- Wikipedia: Taylor series - Comprehensive overview with history and applications.
- Wolfram MathWorld: Taylor Series - Technical details and examples.