What Is a Power Series?

A power series rewrites a function as an infinite polynomial centered at a specific point. To understand what is a power series, consider the definition from OpenStax Calculus Volume 2, Chapter 6: a power series centered at \(a\) has the form \(\) \(\sum_{n=0}^{\infty} c_n (x-a)^n\). Every term is a coefficient \(c_n\) multiplied by \((x-a)^n\), and the sum runs from \(n=0\) to infinity. The series is not magic: it is a limit of its partial sums, and it only equals the function inside its radius of convergence. Outside that radius, the polynomial sum either diverges or gives a value that has nothing to do with the original function. That distinction, the difference between a series that represents the function and a series that lies about it, is what most newcomers miss.

The compact notation is \(\sum_{n=0}^{\infty} c_n (x-a)^n\). The index n starts at zero, so the first term is c_0 (the constant term) and the second term is c_1 (x-a). The center a is the point around which the series is built. If a=0, the series is called a Maclaurin series, which is just a Taylor series centered at zero.

Each coefficient c_n is a real number. In a Taylor series, c_n = f^{(n)}(a) / n!, but a general power series can have any coefficients. The notation box below gives the standard form.

General form: \(\sum_{n=0}^{\infty} c_n (x-a)^n\)
Center: a (a real number)
Coefficients: c_n (real numbers, vary with series)
Partial sum S_N: \(\sum_{n=0}^{N} c_n (x-a)^n\)
Remainder R_N: f(x) - S_N(x)

The center a determines where the series is most accurate. For a series centered at a=0, the approximation is best for x values near zero. For a series centered at a=1, the accuracy peaks near x=1. Changing the center changes the entire series, not just where you plug in.

Coefficients c_n control how much each power contributes. In the geometric series \(\sum x^n\), every coefficient is 1. In the series for e^x, c_n = 1/n!. The coefficients can be found by taking derivatives at the center and dividing by n!, but for many standard functions the coefficients follow a known pattern you can look up.

Every power series has a radius of convergence R, a number (or infinity) that tells you how far from the center the series converges. Inside the open interval (a-R, a+R), the series converges absolutely. Outside, it diverges. At the endpoints x = a \(\pm\) R, you must test separately using the alternating series test, the p-series test, or a comparison test.

The ratio test from OpenStax Chapter 5 gives R = \(\lim\) |c_n / c_{n+1}| when the limit exists. The root test gives R = 1 / \(\lim\)sup |c_n|^{1/n}. For the geometric series \(\sum x^n\), R=1 and the interval is (-1, 1) (both endpoints diverge). For the alternating harmonic series representation of \(\ln(1+x)\), the radius is also 1, but x=1 converges conditionally while x=-1 diverges.

Geometric Series

The simplest power series is the geometric series: \(\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots\) This converges to 1/(1-x) when |x| < 1 and diverges otherwise. The radius is R=1 and the interval is (-1, 1). This series is the foundation for many others because you can substitute, differentiate, or integrate term-by-term to get new representations.

Exponential Series

The Maclaurin series for e^x is \(\sum_{n=0}^{\infty} x^n / n! = 1 + x + x^2/2! + x^3/3! + \cdots\). The radius of convergence is R = \(\infty\), so this series works for every real x. The coefficients follow the pattern c_n = 1/n! because every derivative of e^x at x=0 is 1.

All Taylor series are power series, but not all power series are Taylor series. A Taylor series has a specific formula for its coefficients: c_n = f^{(n)}(a)/n!. A general power series can have any coefficients at all, as long as they are real numbers. The geometric series \(\sum x^n\) is a power series but not a Taylor series of any elementary function (though it matches the Taylor series of 1/(1-x) at a=0). The distinction matters because a Taylor series guarantees that the series equals the function inside the interval of convergence, while an arbitrary power series might not correspond to any function you can write in closed form.

Practical Use

Power series turn complicated functions into polynomials, which you can add, multiply, differentiate, and integrate term-by-term. This is how calculators and computers evaluate functions like \(\sin x\) and e^x: they use a truncated power series (a partial sum) with enough terms to reach the required precision. The Lagrange error bound from OpenStax Chapter 6 gives the maximum error when you stop at a certain term: |R_N(x)| \(\le\) M |x-a|^{N+1} / (N+1)!, where M is the maximum of the (N+1)th derivative on the interval between a and x.

Solving Differential Equations

Power series also solve differential equations that resist exact solutions. Engineers assume a power series solution and calculate the coefficients term by term inside the radius of convergence. The alternating series estimation theorem provides a simpler error bound for alternating series: the error is less than the first omitted term.

Three-Term Approximation

Use the Maclaurin series for e^x: \(\sum_{n=0}^{\infty} x^n / n!\). Plug in x=0.5: 1 + 0.5 + (0.25)/2 = 1 + 0.5 + 0.125 = 1.625. The error is about 0.02372.

Five-Term Approximation

With five terms (n=0 through 4): S_4(0.5) = 1 + 0.5 + 0.125 + 0.020833 + 0.0026042 = 1.64844. The error drops to about 0.00028. Adding more terms inside the radius of convergence reduces the remainder, but outside the radius, more terms make the error worse.