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Taylor's Series Explained

A Taylor series represents a function as an infinite sum of polynomial terms, built from the function's derivatives at a single point.

Vitor Dal Mora3 min read
Contents
  1. Definition
  2. Intuition
  3. How to Compute One
  4. Example: e^x at a = 0
  5. Common Maclaurin Series
  6. Accuracy and Error
  7. Why It Matters
  8. Quick Python Example

Definition #

If f is infinitely differentiable at x = a, its Taylor series centered at a is:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n

Expanded:

f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots

When a = 0, it is called a Maclaurin series:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\,x^n

Intuition #

Each term adds a better local approximation:

  • Order 0: f(a) matches the function's value at a.
  • Order 1: adds the slope, giving the tangent line.
  • Order 2: adds curvature, giving a parabola that hugs the function.
  • Higher orders: keep matching more derivatives, so the polynomial stays close to f over a wider region.

Truncating the series after N terms gives the Taylor polynomial of degree N, a practical approximation.

How to Compute One #

  1. Pick the center a.
  2. Compute the derivatives f'(a), f''(a), f'''(a), \dots
  3. Plug them into \frac{f^{(n)}(a)}{n!}(x - a)^n.
  4. Look for a pattern to write the general term.

Example: e^x at a = 0 #

Every derivative of e^x is e^x, and e^0 = 1, so:

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Common Maclaurin Series #

Function Series Converges for
e^x \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!} all x
\sin x \displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} all x
\cos x \displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} all x
\dfrac{1}{1-x} \displaystyle\sum_{n=0}^{\infty} x^n \lvert x \rvert < 1
\ln(1+x) \displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} -1 < x \le 1

Accuracy and Error #

The error of a degree-N Taylor polynomial T_N(x) is given by the Lagrange remainder:

R_N(x) = \frac{f^{(N+1)}(c)}{(N+1)!}(x - a)^{N+1}, \quad \text{for some } c \text{ between } a \text{ and } x

Rules of thumb:

  • The closer x is to a, the smaller the error.
  • Higher degree usually means a better approximation.
  • The series only equals f inside its radius of convergence.

Why It Matters #

  • Numerical computing: calculators and libraries approximate \sin, \cos, e^x with polynomials.
  • Physics and engineering: the small-angle approximation \sin x \approx x comes from the first term.
  • Machine learning: gradient descent and Newton's method rely on first- and second-order Taylor expansions of the loss.
  • Integration: functions without elementary antiderivatives can be integrated term by term.

Quick Python Example #

import math

def taylor_exp(x, terms=10):
    return sum(x**n / math.factorial(n) for n in range(terms))

print(taylor_exp(1))   # ≈ 2.71828
print(math.exp(1))     # 2.718281828...

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