Functional Series and Uniform Convergence

Why Uniform Convergence is Needed

Let the series Σfₙ(x) converge at each point x to a function S(x). Can we assert that S is continuous if all the fₙ are continuous? Or that the integral of the series equals the series of the integrals?

Answer: no, if the convergence is only pointwise! A stronger form is required — uniform convergence.

Uniform Convergence

A series Σfₙ converges uniformly to S on a set E if for any ε > 0 there exists N such that for all n > N and all x ∈ E: |S(x) - Sₙ(x)| < ε. The same N works for all x.

Cauchy Criterion: The series converges uniformly ⟺ for any ε > 0 there exists N such that for m > n > N: |∑ₖ₌ₙ₊₁ᵐ fₖ(x)| < ε for all x ∈ E.

Weierstrass M-test (majorant test): If |fₙ(x)| ≤ Mₙ for all x and ΣMₙ converges, then Σfₙ converges uniformly.

Properties of Uniformly Convergent Series

If Σfₙ converges uniformly on [a, b] and all fₙ are continuous, then:

• S(x) = Σfₙ(x) is continuous on [a, b]
• Termwise integration is allowed: ∫ₐᵇ S(x)dx = Σ∫ₐᵇ fₙ(x)dx
• With additional conditions — termwise differentiation is allowed

Power Series

A power series: Σcₙ(x-a)ⁿ. This is an “infinite polynomial”.

Abel’s Theorem: There exists a radius of convergence R ∈ [0, +∞] such that the series converges absolutely for |x-a| < R and diverges for |x-a| > R.

Cauchy–Hadamard Formula: 1/R = limsup ⁿ√|cₙ|.

Properties of power series: Inside the circle of convergence, the series can be differentiated and integrated termwise, and the radius of convergence remains the same.

Functions given by convergent power series are called analytic. All elementary functions are analytic in their domains of definition.

Pointwise and Uniform Convergence: A Visual Example

Consider fₙ(x) = xⁿ on [0, 1]. Pointwise: f(x) = 0 for x ∈ [0, 1), f(1) = 1 — a discontinuous function! All fₙ are continuous, but the limit is discontinuous. The convergence is not uniform: sup|fₙ(x) − f(x)| = sup_{x∈[0,1)} xⁿ = 1 does not tend to zero.

This example explains why uniform convergence is the minimal condition under which the limit preserves continuity.

Taylor Series as Power Series

A power series with Taylor coefficients f^(n)(a)/n! and radius of convergence R gives an expansion of an analytic function. For eˣ: R = ∞ (converges everywhere). For 1/(1−x): R = 1 (the series 1 + x + x² + ... diverges for |x| ≥ 1). For 1/(1+x²): R = 1 in the real case, but the cause is a pole at x = ±i in the complex plane.

Functional Series in Data Analysis and Machine Learning

Functional series are a tool of approximation. The method of least squares minimizes ‖f − Sₙ‖² where Sₙ is the partial sum of a series. This produces orthogonal projections in function spaces. In machine learning, neural networks with activations (ReLU, sigmoid) can be viewed as nonlinear “functional series” — superpositions of basis functions with trainable weights.

Weierstrass Approximation Theorem: Any continuous function on a segment can be uniformly approximated by polynomials. This is the basis of polynomial regression.

Testing Uniform Convergence

In practice, the Weierstrass M-test is most convenient. Example: the series Σ sin(nx)/n². Estimate: |sin(nx)/n²| ≤ 1/n², and Σ1/n² = π²/6 converges. Hence, the original series converges uniformly on the whole real line — and its sum is continuous, integrable termwise.

Interchanging the Limit and the Integral under Uniform Convergence

A key consequence of uniform convergence is the lawfulness of termwise integration: if Σfₙ(x) ⇒ S(x) uniformly on [a, b], then ∫ₐᵇ S(x) dx = Σ ∫ₐᵇ fₙ(x) dx. This means the possibility of computing integrals of “complex” functions by integrating simpler terms of the series. Similarly — termwise differentiation requires uniform convergence of the series of derivatives, not of the original series. A practical application: the integral ∫₀¹ ln(1+x)/x dx = Σ(-1)ⁿ/(n+1)² = π²/12 is computed precisely by termwise integration of the series ln(1+x) = x − x²/2 + x³/3 − ...

Question for reflection: Why does a power series inside its circle of convergence always converge uniformly on any strictly smaller circle, but may not converge uniformly on the entire open disk of convergence?

Uniform Convergence and the Continuity of the Sum of a Series

If the series Σuₙ(x) converges uniformly on [a, b] and each uₙ is continuous, then the sum S(x) = Σuₙ(x) is also continuous. Similarly: a uniformly convergent series of integrable functions can be integrated termwise — ∫Σuₙ dx = Σ∫uₙ dx. Weierstrass M-test: if |uₙ(x)| ≤ Mₙ for all x and ΣMₙ converges, then Σuₙ converges uniformly and absolutely. Example: Σxⁿ/n² converges uniformly on [−1, 1] (Mₙ = 1/n²), so the sum is continuous on the entire segment. In function theory, uniform convergence is the condition that allows “interchanging” the limit transition and operations of integration or differentiation.