Sequences and Series of Functions: Convergence Explained

A sequence of functions is a list f1, f2, f3, … where each fn is a function on a common domain, and a series of functions Σ fn is studied through its partial sums SN(x) = f1(x) + … + fN(x). Instead of numbers converging, we now ask whether functions converge — and the answer can depend on x.

This is the language of power series, Fourier series and every infinite expansion used in science. A Taylor series is a series of functions, and asking "does it converge?" really means asking it at every point of an interval, possibly getting different answers at different points.

Key Concepts

• Pointwise convergence: fn → f pointwise on D if for every fixed x in D, the number sequence fn(x) converges to f(x).
• Limit function: the pointwise limit f(x) = limn→∞ fn(x), which may be badly behaved even when every fn is nice.
• Series of functions: Σ fn converges at x when the number series Σ fn(x) converges; its sum is S(x) = limN→∞ SN(x).
• Classic warning: fn(x) = xn on [0, 1] converges pointwise to a function that is 0 on [0, 1) and 1 at x = 1 — discontinuous, though every fn is continuous.
• Radius of convergence: for a power series Σ an(x − c)n, the set of x where it converges is always an interval centred at c.

Formulas

• Pointwise limit: f(x) = limn→∞ fn(x), computed one x at a time.
• Partial sums: SN(x) = Σn=1N fn(x); the series converges at x exactly when the sequence SN(x) converges.
• Geometric series of functions: Σn=0∞ xn = 1/(1 − x) for |x| < 1, diverges for |x| ≥ 1.
• Weierstrass function (preview): Σ ancos(bnπx) converges for 0 < a < 1 to a function continuous everywhere but differentiable nowhere — limits of nice functions can be wild.

Worked Example

Find the pointwise limit of fn(x) = xn on [0, 1].

1. Fix x = 1: fn(1) = 1n = 1 for every n, so the limit at x = 1 is 1.
2. Fix any x with 0 ≤ x < 1: then |x| < 1, and xn → 0 as n → ∞ (a geometric sequence with ratio below 1 in absolute value).
3. Hence the pointwise limit function is f(x) = 0 for 0 ≤ x < 1, and f(1) = 1.
4. Each fn is continuous on [0, 1], but the limit f jumps at x = 1. Pointwise convergence does not preserve continuity.

Common Misconceptions

• "The limit of continuous functions is continuous." The xn example refutes this; you need uniform convergence to preserve continuity.
• "Convergence at each point means the same speed everywhere." Pointwise convergence allows the rate to depend on x — near x = 1, xn converges to 0 agonisingly slowly.
• "You can differentiate term by term." Not in general: Σ xn/n² converges nicely, but naive termwise differentiation needs uniform convergence of the derived series.

Real-World Applications

• Taylor and Maclaurin series in calculators and physics: infinite polynomials approximating sin x, ex and other functions.
• Fourier series in audio and image processing: building signals from sine and cosine functions.
• Solving differential equations by series: assuming a power-series solution and determining its coefficients, valid where the series converges.

Sequences and series of functions converge one point at a time, and the resulting limit function need not inherit the good behaviour of its approximations. The xn example is the one to remember: perfectly continuous approximants, a discontinuous limit. Stronger notions of convergence — uniform convergence and the Weierstrass M-test — were invented precisely to recover the properties that pointwise convergence loses.