Taylor Series: Approximating Functions with Polynomials

Introduction

Imagine you have a complicated function like sin(x) or e^x. Computing these exactly requires advanced mathematical techniques. But what if you could approximate them using simple polynomials?

That's the power of Taylor series — a revolutionary technique in calculus that lets you express almost any smooth function as an infinite polynomial. This simple idea has changed mathematics, physics, engineering, and computer science.

In this post, we'll explore what Taylor series are, how to derive them, and why they matter.

The Core Idea

The main insight: If you know a function's value and all its derivatives at a single point, you can reconstruct the entire function using a polynomial.

This is the foundation of Taylor series.

The Taylor Series Formula

For a function f(x) that's infinitely differentiable at a point a, the Taylor series expansion around a is:

Or more compactly:

Where:

• f^(n)(a) is the n-th derivative of f evaluated at x = a
• n! is n factorial
• (x-a)^n is the power term

Special Case: Maclaurin Series

When a = 0 (expanding around the origin), the Taylor series becomes the Maclaurin series:

Maclaurin series are simpler because you only need derivatives at x = 0.

Intuition: Why Does It Work?

The idea is brilliant in its simplicity:

• Constant term: f(a) — the value at the expansion point
• Linear term: f'(a)(x-a) — approximates how the function changes near a
• Quadratic term: f''(a)(x-a)²/2! — captures curvature
• Higher terms: Capture increasingly fine details

By including more and more terms, you get better and better approximations.

Examples: Common Taylor Series

1. Exponential Function: e^x

2. Sine Function: sin(x)

3. Cosine Function: cos(x)

4. Natural Logarithm: ln(1+x)

5. Geometric Series: 1/(1-x)

Computing a Taylor Series: Step by Step

Let's compute the Taylor series for f(x) = e^x around a = 0:

f(x) = e^x          →  f(0) = 1
f'(x) = e^x         →  f'(0) = 1
f''(x) = e^x        →  f''(0) = 1
f'''(x) = e^x       →  f'''(0) = 1
...

e^x = 1 + 1·x + 1·x²/2! + 1·x³/3! + ...
    = 1 + x + x²/2! + x³/3! + ...

Practical Example: Approximating sin(x)

Let's use Taylor series to approximate sin(π/6) ≈ 0.5:

x = π/6 ≈ 0.5236

0 terms:  sin(x) ≈ 0
1 term:   sin(x) ≈ x ≈ 0.5236
2 terms:  sin(x) ≈ x - x³/6 ≈ 0.4998
3 terms:  sin(x) ≈ x - x³/6 + x⁵/120 ≈ 0.5000
Exact:    sin(π/6) = 0.5

Python Implementation

import math
from math import factorial

def taylor_sin(x, n_terms):
    """Approximate sin(x) using Taylor series with n terms"""
    result = 0
    for n in range(n_terms):
        term = ((-1) ** n) * (x ** (2*n + 1)) / factorial(2*n + 1)
        result += term
    return result

def taylor_cos(x, n_terms):
    """Approximate cos(x) using Taylor series with n terms"""
    result = 0
    for n in range(n_terms):
        term = ((-1) ** n) * (x ** (2*n)) / factorial(2*n)
        result += term
    return result

def taylor_exp(x, n_terms):
    """Approximate e^x using Taylor series with n terms"""
    result = 0
    for n in range(n_terms):
        term = (x ** n) / factorial(n)
        result += term
    return result

# Test
x = math.pi / 6  # π/6 radians

print("Approximating sin(π/6):")
for n in [1, 2, 3, 5, 10]:
    approx = taylor_sin(x, n)
    exact = math.sin(x)
    error = abs(approx - exact)
    print(f"  {n:2d} terms: {approx:.6f}, error: {error:.2e}")

print(f"\nExact: {math.sin(x):.6f}")

# Output:
# Approximating sin(π/6):
#   1 terms: 0.523599, error: 2.36e-02
#   2 terms: 0.499840, error: 1.60e-04
#   3 terms: 0.500005, error: 4.54e-06
#   5 terms: 0.500000, error: 2.57e-11
#  10 terms: 0.500000, error: 0.00e+00
#
# Exact: 0.500000

Convergence: When Does It Work?

Taylor series don't always converge everywhere. The radius of convergence is the region where the series works.

Examples:

• e^x: Converges for all x (radius = ∞)
• sin(x), cos(x): Converge for all x (radius = ∞)
• ln(1+x): Converges for |x| < 1
• 1/(1-x): Converges for |x| < 1

The radius of convergence depends on singularities (points where the function is undefined).

Why Taylor Series Matter

1. Computing Difficult Functions

Computers can't directly compute sin(x) or ln(x). Instead, they use Taylor series to approximate these functions to required precision.

2. Calculus and Analysis

Taylor series bridge discrete approximations and continuous analysis. They're fundamental to numerical analysis.

3. Physics and Engineering

Many physical laws lead to differential equations. Taylor series solve them numerically.

4. Machine Learning

Gradient descent (used in all neural networks) is based on Taylor series approximations of loss functions.

5. Signal Processing

Fourier series (a cousin of Taylor series) are fundamental to digital signal processing.

Taylor vs. Fourier Series

Taylor Series: Approximates using polynomials (powers of x), good for single-point analysis

Fourier Series: Approximates using sines and cosines, good for periodic functions

Error Estimation: How Close Is Our Approximation?

The Lagrange remainder formula tells us the error when truncating a Taylor series:

Where M is an upper bound on |f^(n+1)| between a and x. This tells you how many terms you need for desired accuracy.

Key Insights

Historical Context

Taylor series were developed by Brook Taylor in the early 1700s (first published in 1715). The Maclaurin series, a special case, was popularized by Colin Maclaurin. Though Newton and Gregory had discovered similar ideas earlier, Taylor's general formulation became foundational to calculus.

Conclusion

Taylor series transform complicated functions into manageable polynomials. This simple idea has proven to be one of the most powerful techniques in mathematics, with applications spanning from pure math to engineering to machine learning.

Understanding Taylor series deepens your appreciation for how mathematical approximations work — and why your computer can calculate sin(x) so quickly and accurately.