Sequences and series of functions are crucial in analysis, building on what we've learned about real-valued sequences. They help us understand how functions behave as limits, which is key for many math and physics applications.

Pointwise and uniform convergence are two ways functions can converge. Pointwise is when functions get closer at each point, while uniform means they get closer everywhere at the same rate. This distinction affects properties like continuity and integration.

Pointwise vs Uniform Convergence

Definitions and Key Differences

• Pointwise convergence of a sequence of functions {fn} to a limit function f
  • For every x in the domain and for every ε > 0, there exists an N (depending on x and ε) such that |fn(x) - f(x)| < ε for all n ≥ N
  • N may vary with x (different N for each x)
• Uniform convergence of a sequence of functions {fn} to a limit function f
  • For every ε > 0, there exists an N (depending only on ε) such that |fn(x) - f(x)| < ε for all x in the domain and all n ≥ N
  • N is independent of x (same N for all x)
• Main distinction between pointwise and uniform convergence
  • Dependence of N on x
    • Pointwise convergence: N may vary with x
    • Uniform convergence: N is independent of x

Implications and Examples

• Uniform convergence implies pointwise convergence, but the converse is not true in general
  • If a sequence of functions converges uniformly, it also converges pointwise to the same limit function
  • Counterexample: fn(x) = xⁿ on [0, 1] converges pointwise but not uniformly
• Uniform convergence preserves certain properties of the limit function under appropriate conditions
  • Continuity
  • Integrability
  • Differentiability
• Examples of sequences with different convergence properties
  • fn(x) = 1/n converges uniformly to 0 on any bounded interval
  • fn(x) = sin(nx)/n converges pointwise to 0 on ℝ but not uniformly

Proving Convergence of Function Sequences

Pointwise Convergence

• To prove pointwise convergence
  • Show that for every x in the domain and ε > 0, there exists an N (possibly depending on x and ε) such that |fn(x) - f(x)| < ε for all n ≥ N
  • Approach: Fix x, find N that works for the given ε
• To disprove pointwise convergence
  • Find an x in the domain and an ε > 0 such that for every N, there exists an n ≥ N with |fn(x) - f(x)| ≥ ε
  • Approach: Fix x, show that no N works for some ε

Uniform Convergence

• To prove uniform convergence
  • Show that for every ε > 0, there exists an N (independent of x) such that |fn(x) - f(x)| < ε for all x in the domain and all n ≥ N
  • Approach: Find N that works for all x simultaneously
• To disprove uniform convergence
  • Find an ε > 0 such that for every N, there exist an x in the domain and an n ≥ N with |fn(x) - f(x)| ≥ ε
  • Approach: Show that no N works for all x simultaneously
• Common techniques for proving or disproving convergence
  • Definition
  • Cauchy criterion
  • Weierstrass M-test

Relationship Between Convergence Types

• Uniform convergence implies pointwise convergence
  • If {fn} converges uniformly to f, then {fn} converges pointwise to f
  • Uniform convergence is a stronger condition than pointwise convergence
• Pointwise convergence does not imply uniform convergence
  • There exist sequences of functions that converge pointwise but not uniformly
  • Example: fn(x) = xⁿ on [0, 1] converges pointwise to f(x) = 0 for x ∈ [0, 1) and f(1) = 1, but not uniformly
• Uniform convergence preserves certain properties of the limit function
  • Continuity: If {fn} is a sequence of continuous functions converging uniformly to f, then f is continuous
  • Integrability: If {fn} is a sequence of integrable functions converging uniformly to f, then f is integrable and lim(n→∞) ∫fn = ∫f
  • Differentiability: If {fn} is a sequence of differentiable functions and {fn'} converges uniformly to g, then f is differentiable and f' = g

Identifying Limit Functions

Pointwise Convergent Sequences

• The limit function f of a pointwise convergent sequence {fn} is defined by f(x) = lim(n→∞) fn(x) for each x in the domain
  • Evaluate the limit of fn(x) as n → ∞ for each x
  • The limit may depend on x
• The limit function of a pointwise convergent sequence may not inherit properties from the sequence
  • Continuity, differentiability, or integrability may not be preserved
  • Example: fn(x) = xⁿ on [0, 1] converges pointwise to a discontinuous function

Uniformly Convergent Sequences

• The limit function f of a uniformly convergent sequence {fn} is also given by f(x) = lim(n→∞) fn(x) for each x in the domain
  • Evaluate the limit of fn(x) as n → ∞ for each x
  • The limit is independent of x
• The limit function of a uniformly convergent sequence inherits properties from the sequence under appropriate conditions
  • Continuity, differentiability, and integrability are preserved
  • Example: fn(x) = 1/n converges uniformly to f(x) = 0, which is continuous, differentiable, and integrable
• In some cases, the limit function may have a closed-form expression
  • Polynomial, exponential, or trigonometric function
  • Example: fn(x) = (1 + x/n)ⁿ converges uniformly to f(x) = eˣ on any bounded interval