Difficulty: Intermediate to Advanced | Prerequisites: Infinite series, convergence and divergence, basic integration.
Power series and Fourier series are two ways of representing functions as infinite sums. A power series writes a function as a sum of polynomial terms (a₀ + a₁x + a₂x² + ...). A Fourier series writes a periodic function as a sum of sines and cosines. Both are tools for approximating, analysing, and computing with functions that are otherwise difficult to work with directly. These topics sit at the boundary of Calculus 1 and more advanced courses.
Power series
An infinite series of the form Σ aₙxⁿ (n = 0, 1, 2, ...), where aₙ are the coefficients and x is a variable. Think of it as an "infinite polynomial" that can represent functions like eˣ, sin x, and ln(1 + x) within a certain range of x values.
Fourier series
A representation of a periodic (wave-like) function as the sum of simple sine and cosine waves of different frequencies. More formally, it decomposes any periodic function into a (possibly infinite) set of oscillating components. Think of it as breaking a complex sound wave into its individual pure tones.
A power series centred at c has the form Σ aₙ(x - c)ⁿ. When c = 0, this is a Maclaurin series.
Every power series has a radius of convergence R: it converges for |x - c| < R and diverges for |x - c| > R. Behaviour at the endpoints |x - c| = R must be checked separately.
Common power series to know:
eˣ = 1 + x + x²/2! + x³/3! + ... (converges for all x)
sin x = x - x³/3! + x⁵/5! - ... (converges for all x)
1/(1 - x) = 1 + x + x² + x³ + ... (converges for |x| < 1, the geometric series)
Power series can be differentiated and integrated term by term within their radius of convergence.
Taylor's theorem provides the coefficients: aₙ = f⁽ⁿ⁾(c) / n!
A Fourier series represents a periodic function f(x) with period 2L as a₀/2 + Σ [aₙ cos(nπx/L) + bₙ sin(nπx/L)].
The coefficients aₙ and bₙ are computed by integration, using the orthogonality of sine and cosine functions.
Fourier series can represent discontinuous functions (such as a square wave), which power series cannot.
The more terms you include, the better the approximation. Near a discontinuity, the partial sums overshoot (the Gibbs phenomenon).
Fourier series are the foundation of signal processing, acoustics, and solving partial differential equations. In a Calculus 1 course, they typically appear as a preview or enrichment topic.
General power series:
\sum_{n=0}^{\infty} a_n (x - c)^nTaylor series coefficients:
a_n = \frac{f^{(n)}(c)}{n!}Fourier series (period 2L):
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]Fourier coefficients:
a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dxb_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dxPower series let calculators and computers evaluate functions like sin x and eˣ by adding up polynomial terms. Every time your calculator computes a trig function, it is using a truncated power series.
Fourier series are the mathematical backbone of audio processing, image compression (JPEG), telecommunications, and any technology that works with signals or waves.
Students sometimes assume a power series converges everywhere. It does not: every power series has a radius of convergence, and outside that radius the series diverges.
Fourier series can represent discontinuous functions, which surprises students who expect all infinite sums of smooth functions (sines, cosines) to be smooth. The Gibbs phenomenon, where partial sums overshoot near a discontinuity, is a classic follow-up exam question.
Students sometimes confuse Taylor series with power series. A Taylor series is a specific kind of power series where the coefficients come from the derivatives of a known function. A general power series need not arise from a Taylor expansion.
⚠️ Memorise the Maclaurin series for eˣ, sin x, cos x, and 1/(1 - x). These appear constantly in exam questions, either directly or as building blocks.
⚠️ Know how to find the radius of convergence using the ratio test.
⚠️ Fourier series may appear as a conceptual question in MATH 104 ("what does it mean to represent a function as a sum of sines and cosines?") rather than a full computation. Know the idea, even if the heavy computation belongs to later courses.
The Maclaurin series for eˣ begins 1 + x + x²/2! + _______. (x³/3!)
True or false: every power series converges for all values of x. _______ (False. It depends on the radius of convergence.)
A Fourier series represents a function as a sum of _______ and _______. (sines, cosines)
The radius of convergence of the geometric series 1 + x + x² + ... is _______. (1, meaning |x| < 1)
True or false: Fourier series can only represent smooth, continuous functions. _______ (False. They can represent discontinuous functions.)
Q: Write the first four non-zero terms of the Maclaurin series for sin x.
A: sin x = x - x³/3! + x⁵/5! - x⁷/7! = x - x³/6 + x⁵/120 - x⁷/5040.
Q: What is the radius of convergence of Σ xⁿ/n! (the series for eˣ)?
A: The radius of convergence is infinite. The ratio test gives |x|/(n+1), which approaches 0 for any fixed x as n grows. The series converges for all real x.
Q: In plain language, what does a Fourier series do?
A: It breaks a periodic function into a recipe of pure sine and cosine waves of different frequencies and amplitudes. Adding these waves together reconstructs the original function.
Q: What is the Gibbs phenomenon?
A: When a Fourier series approximates a function with a discontinuity, the partial sums overshoot near the jump by roughly 9%, no matter how many terms you include. The overshoot narrows but does not disappear.
Power series connect to Taylor's theorem (the source of the coefficients), convergence tests (ratio test, root test), and the representation of functions that are difficult to integrate in closed form. Fourier series connect to differential equations, signal processing, and linear algebra (orthogonality of functions). Both topics extend the idea of infinite series from the first study notes document into powerful tools for representing and analysing functions.
Power series, Taylor series, Maclaurin series, radius of convergence, interval of convergence, ratio test, root test, term-by-term differentiation, term-by-term integration, Fourier series, Fourier coefficients, orthogonality, Gibbs phenomenon, periodic functions, sine, cosine, complex exponentials, signal processing, MATH 104, Calculus 1, Penn State, advanced calculus series