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🏃🏽‍♀️‍➡️Intro to Mathematical Analysis

Uniform convergence is a powerful concept in mathematical analysis. It allows us to understand how sequences of functions behave and what properties they maintain as they converge to a limit function.

Continuity and differentiation of uniformly convergent series are key applications of this concept. These properties help us analyze complex functions and solve problems in calculus and real analysis.

Continuity of Uniform Limits

Uniform Convergence and Continuity

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  • A sequence of functions {fn} converges uniformly to a function f on a set A when for every ε > 0, there exists an N ∈ ℕ such that |fn(x) - f(x)| < ε for all n ≥ N and all x ∈ A
  • The uniform convergence of a sequence of functions {fn} to a limit function f on a set A implies that the distance between fn and f becomes arbitrarily small for all x ∈ A as n increases
  • If each function fn in a sequence {fn} is continuous on a set A and the sequence converges uniformly to a function f on A, then the limit function f is also continuous on A (uniform limit theorem)
  • The uniform limit theorem establishes a connection between the continuity of the individual functions fn and the continuity of the limit function f, provided that the convergence is uniform

Proving the Continuity of Uniform Limits

  • The proof of the uniform limit theorem involves using the definition of continuity and the properties of uniform convergence to show that for any x0 ∈ A and ε > 0, there exists a δ > 0 such that |x - x0| < δ implies |f(x) - f(x0)| < ε
  • The key steps in the proof are choosing an appropriate N based on ε/3, using the continuity of fN to find a δ, and applying the triangle inequality to show that |f(x) - f(x0)| < ε
  • The triangle inequality |f(x) - f(x0)| ≤ |f(x) - fN(x)| + |fN(x) - fN(x0)| + |fN(x0) - f(x0)| is used to break down the difference |f(x) - f(x0)| into manageable parts
  • By carefully selecting N and δ, each part of the triangle inequality can be made smaller than ε/3, ensuring that |f(x) - f(x0)| < ε whenever |x - x0| < δ, thus proving the continuity of f

Continuity of Uniformly Convergent Series

Uniform Convergence of Series and Continuity

  • If a series of functions Σ∞n=1 fn(x) converges uniformly to a function f on a set A, and each function fn is continuous on A, then the sum function f is also continuous on A
  • The continuity of a function defined by a uniformly convergent series can be established by applying the uniform limit theorem to the sequence of partial sums {Sn(x)} of the series, where Sn(x) = Σnk=1 fk(x)
  • The partial sums {Sn(x)} form a sequence of continuous functions that converge uniformly to the sum function f, allowing the application of the uniform limit theorem to conclude that f is continuous on A

Weierstrass M-Test for Uniform Convergence

  • The Weierstrass M-test is a useful tool for establishing the uniform convergence of a series of functions, especially when the functions fn are bounded by a convergent series of constants Mn
  • If |fn(x)| ≤ Mn for all x ∈ A and all n ∈ ℕ, and the series Σ∞n=1 Mn converges, then the series Σ∞n=1 fn(x) converges uniformly on A
  • The Weierstrass M-test simplifies the process of verifying uniform convergence by comparing the functions fn to a series of constants Mn, which is often easier to work with than the original series
  • Examples of series that can be shown to converge uniformly using the Weierstrass M-test include geometric series (Σ∞n=1 ar^n, |r| < 1) and certain power series (Σ∞n=0 an(x - c)^n) on compact subintervals of their intervals of convergence

Term-by-Term Differentiation of Series

Uniform Convergence and Differentiation

  • If a series of functions Σ∞n=1 fn(x) converges uniformly on an interval I, and each function fn is differentiable on I, the series of derivatives Σ∞n=1 f'n(x) may not necessarily converge to the derivative of the sum function f
  • The uniform convergence of the original series Σ∞n=1 fn(x) does not guarantee the uniform convergence of the series of derivatives Σ∞n=1 f'n(x) or the differentiability of the sum function f
  • For a uniformly convergent series Σ∞n=1 fn(x) to be differentiable term by term on an interval I, the series of derivatives Σ∞n=1 f'n(x) must also converge uniformly on I

Sufficient Conditions for Term-by-Term Differentiation

  • If both the original series Σ∞n=1 fn(x) and the series of derivatives Σ∞n=1 f'n(x) converge uniformly on an interval I, then the sum function f(x) = Σ∞n=1 fn(x) is differentiable on I, and its derivative is given by f'(x) = Σ∞n=1 f'n(x)
  • The uniform convergence of the series of derivatives is a sufficient condition for term-by-term differentiation, as it ensures that the limit of the derivatives of the partial sums converges to the derivative of the sum function
  • In some cases, the original series may be differentiable term by term even if the series of derivatives does not converge uniformly, but additional analysis is required to establish this result
  • Power series (Σ∞n=0 an(x - c)^n) are examples of series that can be differentiated term by term within their intervals of convergence, as they converge uniformly on compact subintervals of their intervals of convergence

Applications of Uniform Convergence Theorems

Determining Uniform Convergence

  • When given a series of functions, first determine whether the series converges uniformly on the given interval or set using tests such as the Weierstrass M-test or by directly verifying the definition of uniform convergence
  • If the series converges uniformly and each function in the series is continuous, conclude that the sum function is continuous on the given interval or set
  • To determine whether a uniformly convergent series can be differentiated term by term, check if the series of derivatives also converges uniformly on the given interval. If so, the sum function is differentiable, and its derivative is the sum of the derivatives of the individual terms

Applying Theorems to Power Series

  • Power series (Σ∞n=0 an(x - c)^n) are important examples of series that can be analyzed using the theorems on continuity and differentiation of uniformly convergent series
  • A power series converges uniformly on any compact subinterval of its interval of convergence, allowing for the application of the uniform limit theorem and term-by-term differentiation within the interval of convergence
  • The continuity and differentiability of functions defined by power series can be established using the properties of uniform convergence, making power series a valuable tool in analysis

Manipulating Series and Functions

  • In some cases, it may be necessary to manipulate the given series or function to fit the form of a known uniformly convergent series, such as a geometric series or a power series, to apply the relevant theorems and properties
  • Techniques such as substitution, term-by-term integration, or term-by-term differentiation can be used to transform a given series into a more manageable form
  • Recognizing the connections between the given series or function and well-known uniformly convergent series can simplify the process of applying the theorems on continuity and differentiation of uniformly convergent series


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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.