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Series Expansion — Definition, Formula & Examples

A series expansion is a way of rewriting a function as an infinite sum of terms, typically powers of a variable. The most common type is the Taylor series, which builds the sum from the function's derivatives at a chosen point.

A series expansion of a function f(x)f(x) about a point x=ax = a is a representation of the form f(x)=∑n=0∞cn(x−a)nf(x) = \sum_{n=0}^{\infty} c_n (x - a)^n, where the coefficients cnc_n are determined by the type of expansion. For a Taylor series, cn=f(n)(a)n!c_n = \frac{f^{(n)}(a)}{n!}. The expansion is valid within the interval of convergence around aa.

Key Formula

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n
Where:
  • f(n)(a)f^{(n)}(a) = The nth derivative of f evaluated at the center point a
  • n!n! = n factorial, the product of all positive integers up to n
  • aa = The center point of the expansion
  • xx = The variable; the expansion converges for x within the radius of convergence

How It Works

To build a series expansion, you pick a center point aa and compute successive derivatives of ff at that point. Each derivative produces one coefficient in the sum. The zeroth term captures the function's value, the first term captures its slope, the second captures its curvature, and so on. Adding more terms improves the approximation near aa. A Maclaurin series is simply a Taylor series centered at a=0a = 0.

Worked Example

Problem: Find the Maclaurin series expansion of exe^x up to the first four terms.
Step 1: Identify the function and center. Here f(x)=exf(x) = e^x and a=0a = 0.
Step 2: Compute derivatives at a=0a = 0. Every derivative of exe^x is exe^x, so f(n)(0)=e0=1f^{(n)}(0) = e^0 = 1 for all nn.
Step 3: Substitute into the Taylor series formula for n=0,1,2,3n = 0, 1, 2, 3.
ex≈10!+11!x+12!x2+13!x3=1+x+x22+x36e^x \approx \frac{1}{0!} + \frac{1}{1!}x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}
Answer: The first four terms of the Maclaurin series for exe^x are 1+x+x22+x361 + x + \frac{x^2}{2} + \frac{x^3}{6}.

Why It Matters

Series expansions let physicists and engineers approximate complex functions with polynomials, which are far easier to compute and integrate. In numerical methods, truncated series underpin algorithms for everything from solving differential equations to rendering computer graphics.

Common Mistakes

Mistake: Using the expansion outside its radius of convergence and expecting accurate results.
Correction: Every series expansion has a radius of convergence RR. The expansion is only valid for ∣x−a∣<R|x - a| < R. Always determine RR using a convergence test before relying on the approximation.