11.4 Sequences and Series of Functions: Pointwise and Uniform Convergence

11.4 Sequences and Series of Functions: Pointwise and Uniform Convergence

Convergence of function sequences is a central topic on the GRE Mathematics Subject Test. Success requires recognizing when limits, integrals, and derivatives can be legitimately interchanged with infinite sequences and series.

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Pointwise vs. Uniform Convergence

Let $(f_n)_{n=1}^\infty$ be a sequence of functions on $E \subseteq \mathbb{R}$.

• Pointwise Convergence: $f_n \to f$ pointwise if for every $x \in E$ and $\varepsilon > 0$, there exists $N(\varepsilon, x)$ such that $n > N \implies |f_n(x) - f(x)| < \varepsilon$.
• Uniform Convergence: $f_n \rightrightarrows f$ uniformly if for every $\varepsilon > 0$, there exists $N(\varepsilon)$ such that $n > N \implies |f_n(x) - f(x)| < \varepsilon$ for all $x \in E$.
  • Supremum Norm: $f_n \rightrightarrows f$ uniformly on $E \iff \lim_{n \to \infty} \|f_n - f\|\infty = \lim{n \to \infty} \sup_{x \in E} |f_n(x) - f(x)| = 0$.

Canonical Diagnostic Examples

1. $f_n(x) = x^n$ on $[0, 1]$: Pointwise limit is $f(x) = 0$ for $x \in [0, 1)$ and $f(1) = 1$. The limit is discontinuous at $x = 1$, so convergence is not uniform on $[0, 1]$. On $[0, b]$ with $b < 1$, $\|f_n\|_\infty = b^n \to 0$, so convergence is uniform.
2. $f_n(x) = \frac{nx}{1 + n^2 x^2}$ on $[0, 1]$: Pointwise limit is $0$ everywhere. The peak occurs at $x = 1/n$ where $f_n(1/n) = 1/2$. Thus $\|f_n - 0\|_\infty = 1/2 \not\to 0$. The limit is continuous, yet convergence is not uniform!

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Uniform Limit Theorem and Dini's Theorem

• Uniform Limit Theorem: If $f_n: E \to \mathbb{R}$ are continuous and $f_n \rightrightarrows f$ uniformly on $E$, then $f$ is continuous on $E$.
  • GRE Test: If continuous functions converge to a discontinuous limit, uniform convergence fails.
• Dini's Theorem: Let $K$ be a compact metric space. If $f_n: K \to \mathbb{R}$ are continuous, $f_n \to f$ pointwise to a continuous limit $f$, and the sequence is monotonic ($f_{n+1}(x) \le f_n(x)$ for all $x$), then $f_n \rightrightarrows f$ uniformly on $K$.

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Interchange of Limits: Integration and Differentiation

Integration Interchange

If $f_n \rightrightarrows f$ uniformly on $[a, b]$ with each $f_n$ Riemann integrable, then $f$ is Riemann integrable and: $\\lim_{n \\to \\infty} \\int_a^b f_n(x) \\, dx = \\int_a^b \\left( \\lim_{n \\to \\infty} f_n(x) \\right) \\, dx = \\int_a^b f(x) \\, dx.$

• Counterexample without uniform convergence: $f_n(x) = 2n^2 x e^{-n^2 x^2}$ on $[0, 1]$ converges pointwise to $0$, but $\int_0^1 f_n(x) \, dx = 1 - e^{-n^2} \to 1 \neq \int_0^1 0 \, dx = 0$.

Differentiation Interchange (The Classic GRE Trap)

Uniform convergence of $f_n$ does not imply $f_n' \to f'$.

• Theorem: Suppose each $f_n$ is continuously differentiable on $[a, b]$, $f_n(x_0)$ converges for at least one $x_0 \in [a, b]$, and the derivatives converge uniformly: $f_n' \rightrightarrows g$ on $[a, b]$. Then $f_n \rightrightarrows f$ uniformly, $f$ is differentiable, and $f'(x) = g(x) = \lim f_n'(x)$.
• Counterexample: $f_n(x) = \frac{\sin(n^2 x)}{n} \rightrightarrows 0$ uniformly on $\mathbb{R}$, but $f_n'(x) = n \cos(n^2 x)$ diverges unboundedly.

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Weierstrass M-Test for Series of Functions

For a series of functions $\sum_{n=1}^\infty u_n(x)$ on $E$:

• Theorem: If there exist constants $M_n \ge 0$ such that $|u_n(x)| \le M_n$ for all $x \in E$ and $\sum_{n=1}^\infty M_n < \infty$, then $\sum_{n=1}^\infty u_n(x)$ converges absolutely and uniformly on $E$.
• Consequences: If each $u_n$ is continuous, the sum $S(x)$ is continuous, and integration can be performed term by term: $\int_a^b S(x) \, dx = \sum_{n=1}^\infty \int_a^b u_n(x) \, dx$.

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Equicontinuity and the Arzelà-Ascoli Theorem

• Equicontinuity: A family $\mathcal{F} \subset C(X)$ is equicontinuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $d(x, y) < \delta \implies |f(x) - f(y)| < \varepsilon$ for all $f \in \mathcal{F}$ simultaneously.
• Arzelà-Ascoli Theorem: Let $K$ be a compact metric space. A subset $\mathcal{F} \subset C(K)$ has compact closure in the supremum norm (every sequence in $\mathcal{F}$ has a uniformly convergent subsequence) if and only if $\mathcal{F}$ is pointwise bounded and equicontinuous.

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Comparison Table: Conditions for Interchanging Operations

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Step-by-Step Worked Problems

Problem 1: Applying the Weierstrass M-Test

Prove that $S(x) = \sum_{n=1}^\infty \frac{\cos(nx)}{n^2 + x^2}$ is continuous on $\mathbb{R}$.

Solution:

1. For all $x \in \mathbb{R}$, $|u_n(x)| = \left| \frac{\cos(nx)}{n^2 + x^2} \right| \le \frac{1}{n^2} = M_n$.
2. The series $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} < \infty$ converges.
3. By the Weierstrass M-Test, the series converges uniformly on $\mathbb{R}$.
4. Since each term is continuous, the Uniform Limit Theorem guarantees $S(x)$ is continuous on $\mathbb{R}$.

Problem 2: Analyzing Derivative Limits

Let $f_n(x) = \frac{x}{1 + nx^2}$ on $[-1, 1]$. Does $f_n \rightrightarrows 0$ uniformly? Does $f_n'(0) \to 0$?

Solution:

1. Maximum occurs at $x = 1/\sqrt{n}$, with $f_n(1/\sqrt{n}) = \frac{1}{2\sqrt{n}} \to 0$.
2. Since $\|f_n - 0\|_\infty = \frac{1}{2\sqrt{n}} \to 0$, $f_n \rightrightarrows 0$ uniformly on $[-1, 1]$.
3. Differentiating: $f_n'(x) = \frac{1 - nx^2}{(1 + nx^2)^2}$, so $f_n'(0) = 1$ for all $n$.
4. Thus $\lim_{n \to \infty} f_n'(0) = 1 \neq 0 = (\lim_{n \to \infty} f_n)'(0)$, showing uniform convergence of $f_n$ is insufficient for derivative interchange.

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GRE Exam Traps & Pitfalls

Trap 1: Interchanging Differentiation Assuming $f_n$ Converges Uniformly The most frequent GRE analysis trap. Uniform convergence of $f_n$ does not imply convergence of $f_n'$. You must establish uniform convergence of the derivative sequence $f_n'$.

Trap 2: Assuming Continuous Pointwise Limits Imply Uniform Convergence If continuous $f_n$ converge to a continuous limit $f$, convergence is not necessarily uniform. For $f_n(x) = \frac{nx}{1 + n^2 x^2}$, the limit is $0$ everywhere but $\|f_n\|_\infty = 1/2$.

Trap 3: Omitting Monotonicity in Dini's Theorem Dini's Theorem requires compact domain, continuous functions with continuous limit, and monotonic convergence. Without monotonicity, uniform convergence can easily fail.