As we already know how to define and write sequences and series of real numbers in mathematics. It is possible to define the sequence and series for functions, i.e., for real values functions. In this article, you will learn how to write the sequences and series of functions and the convergence of sequence and series of functions. Also, get the solved problems on sequences and series of functions here.
Learn in detail about sequences and series.
Sequences of Functions
Suppose fn be the real-valued function defined on E ⊆ R for each n ∈ N, then the set {fn}; n = 1, 2, 3, 4,…. is called the sequence of real-valued functions on E. It is denoted by {fn}, <fn> or {fn : E → R, n ∈ N.
Example:
If fn(x) = xn; x ∈ [0, 1] is a real valued function then the sequence of real valued functions is written as: {fn} = {f1(x), f2(x), f3(x),….} = {x, x2, x3,….} defined on [0, 1].
Series of Functions
Suppose {fn} be the sequence of real-values functions defined by E ⊆ R such that the expression
Example:
Let fn(x) = (cos nx)/n2 defined on [0, 1] then the series of functions is given by:
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Convergence of Sequences and Series of Functions
Now, you will learn about the convergence of series and sequence of functions. Let’s recall what is the convergence of a sequence and convergence of a series.
A sequence {an} is said to be convergent, if limn→∞ (an) = finite.
A series is convergent when it approaches a specific value as the series approaches infinity. It is denoted by
Convergence of Sequence of Functions
When coming to the convergence of sequence and series of functions, we can define pointwise and uniform convergence.
Pointwise Convergence:
A sequence of functions fn: X → ℝ, where X is a subset of ℝ, 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),….
Uniform convergence
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 → ∞.
Convergence of Series of Functions
Pointwise convergence:
A series of functions
Uniform convergence:
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
Solved Example
State and prove Cauchy’s criterion for uniform convergence of sequence of functions.
Statement:
A sequence of functions {fn} is defined on E converges uniformly on E if and only if given ε > 0 there exists m ∈ N such that |fn+p (x) – fn(x)| < ε for all n ≥ m, p ≥ 1 and x ∈ E.
Proof:
Let us assume that the sequence of function {fn} defined on E converges uniformly to limit function f.
Therefore, for ε > 0 there exists m ∈ N such that
|fn(x) – f(x)| < ε/2 for all n ≥ m and x ∈ E……(1)
From the given statement, n ≥ m, p ≥ 1 and x ∈ E
So, |fn+p (x) – f(x)| < ε/2……(2)
Now,
|fn+p (x) – fn(x)| = |fn+p(x) – f(x) + f(x) – fn(x)|
≤ |fn+p(x) – f(x)| + |f(x) – fn(x)|
≤ |fn+p(x) – f(x)| + |fn(x) – f(x)|
< (ε/2) + (ε/2) [from (1) and (2)]
< ε
Thus, |fn+p (x) – fn(x)| < ε for all n ≥ m, p ≥ 1 and x ∈ E.
Hence proved.
Problems on Sequences and Series of Functions
- Show that the sequence of functions {fn} where fn(x) = xn/n defined on [0, 1] converges uniformly to 0.
- Show that the sequence of functions, {fn}, where fn(x) = tan-1(nx), x ≥ 0 converges uniformly on [a, b], a > 0 but is only pointwise convergent on [0, b].
- Show that x = 0 is a point of non-uniform convergence of the series x2 + [x2/(1 + x2)] + [x2/(1 + x2)2] + ……
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