Before learning about uniform convergence, let us recall and understand a few related terms and concepts, such as sequence, series, the convergence of a sequence, point convergence, etc. Learning these terms will help you to understand the uniform convergence quickly.
Sequence:
An ordered collection of objects in which repetitions are allowed. It comprises members, called elements or terms like a set.
Series:
The cumulative sum of a given sequence of terms.
Learn in detail about Sequence and series here.
Convergence of a sequence:
A sequence {an} is said to be convergent, if limn→∞ (an) = finite. An example of a convergence sequence is given below.
Let {an} = 1/2n, for all n ∈ N
So, limn→∞ (an) = limn→∞ (1/2n) = 1/2∞ = 1/∞ = 0 = finite
Sequence of functions:
A set of functions fn(x), n = 1, 2,… defined on a common domain D.
Pointwise Convergence:
A sequence of functions fn: X → ℝ, where X is a subset of ℝ, is said to be converges pointwise on X to the function f: X → ℝ if and only if
limn→∞ fn(x) = f(x) ∀ x ∈ R
Here, fn(x) = f1(x), f2(x), f3(x),….
In the same way, a series of functions
What is Uniform Convergence?
Uniform convergence can be defined for both sequences of functions and series of functions, as given below.
Uniform Convergence of Sequence
A sequence of functions fn(x); n = 1, 2, 3,…. Is said to be uniformly convergent to f for a set E of values of x, if for each ε > 0, a positive integer N exists such that |fn(x) – f(x)| < ε for n ≥ N and x ∈ E.
Alternatively, we can define the uniform convergence of a sequence of functions, as follows.
A sequence of functions fn(x); n = 1, 2, 3,…. Is said to be converges uniformly to f if and only if;
That means, supx∈E |fn(x) – f(x)| → 0 as n → ∞.
Uniform Convergence of Series
A series of functions ∑fn(x); n = 1, 2, 3,… is said to be uniformly convergent on E if the sequence {Sn} of partial sums defined by
Alternatively, we can define the uniform convergence of a series as follows.
Suppose gn(x) : E → ℝ is a sequence of functions, we can say that the series
Below are simple examples of uniform convergence.
For x ∈ [0, 1), the sequence (1/2)x+n converges uniformly
For x ∈ [0, 1), the sequence xn does not converge uniformly
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Let’s understand this uniform convergence concept in a better way with the help of the solved problems given below.
Uniform Convergence Solved Examples
Example 1:
Show that the sequence of function {fn}, where fn(x) = 1/(x + n) is uniformly convergent in any interval [0, b], b > 0.
Solution:
Given,
fn(x) = 1/(x + n)
Let’s apply the limit for this function.
f(x) = limn→∞ fn(x) = 0 ∀ x ∈ [0, b]
That means the sequence converges pointwise to 0.
For any ε > 0,
|fn(x) − f(x)| = 1/(x + n) < ε if n > (1/ε) − x, which decreases with x, and 1/ε will be the maximum value.
Let N be an integer such that N ≥ 1/ε
Thus, for ε > 0, there exists N such that |fn(x) − f(x)| < ε, for all n ≥ N
Hence, the sequence of function fn(x) = 1/(x + n) is uniformly convergent in any interval [0, b], b > 0.
Example 2:
Prove that xn is not uniformly convergent.
Solution:
Consider the sequence of functions {xn} defined on [0, 1].
Thus, we quickly identified the pointwise limit of this function.
Indeed, when x ∈ (0, 1), xn → 0 as n → ∞ and, when x = 1, xn → 1 as n → ∞.
Here, we can observe that the pointwise limit of the given sequence is the function ψ(x) = 0, x ∈ [0, 1) and ψ(1) = 1.
So, we say that this sequence is not uniform convergent.
For x ∈ [0, 1), xn = |xn − 0| < ε if and only if n > log ε/ log x such that n0(x) > log ε/ log x.
As x comes close to 1, n0(x) becomes unbounded.
Consequently, there is no way to identify an n0 to make |xn − 0| < ε, n ≥ n0, ∀ x ∈ [0, 1).
Therefore, xn is not uniformly convergent.
Hence proved.
Practice Problems
- Prove that the sequence {fn}, where fn(x) = x/(1 + nx2), x is real, converges uniformly on any closed interval I.
- Find the uniform convergence of fn(x) = ex/n and gn(x) = xn on [0, 1].
- Prove that the sequence {fn}, where fn(x) = xn−1 (1 −x) converges uniformly in the interval [0, 1].
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