11.1 Pointwise and Uniform Convergence of Sequences
4 min read•Last Updated on July 30, 2024
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
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Top images from around the web for Definitions and Key Differences
Autonomous convergence theorem - Wikipedia, the free encyclopedia View original
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Wijsman Rough λ Statistical Convergence of Order α of Triple Sequence of Functions View original
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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