Calculus How To

Sequence of Functions, Limit Function

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Series and Sequences >

A sequence of functions {fn(x)} is a family of functions where “x” (or the parameter set) is a sequence of natural numbers (the counting numbers 1, 2, 3…).

For example, let’s say the sequence of functions was defined by {xn}. Each n will result in a new function:

  • x1,
  • x2,
  • x3,
  • …xn.

You can also fix the x to get a numeric sequence; If x = 2 then the formula becomes the geometric sequence {2n [1].

Convergence of Sequence of Functions

If you a working with real-valued functions, a good place to start studying convergence is to graph all of the functions (substituting in 1, 2, 3, … for “n”) to see where they converge.

For example, the sequence of functions f(x) = x/n converges pointwise to the limit function f(x) = 0 for the closed interval [0, 1]:

family of functions pointwise convergence example

This series of functions converges pointwise to f(x) = 0.

Pointwise convergence is defined as follows [2]:

A sequence {fn} of functions fn: X → ℝ converges pointwise to a function fn: X → ℝ if for every x the sequence of real numbers {fn(x)} converges the the number f(x).

Uniform convergence is defined more strictly than pointwise convergence. Values must stay in a “box” around the function.

uniform convergence 2

A uniformly convergent function, f(x), surrounded by an ε strip. An integer N is inside the strip for all x in the domain.

Proving Convergence of a Sequence of Functions

Showing that a sequence of functions converges is relatively simple. You can graph the possibilities and see where they converge. Or, plug in a few values in the given interval and see where the functions head that way. Proving that a sequence of functions converges is a little trickier. It’s not difficult, but as it’s a formal proof, it involves a fair amount of terminology like epsilon(ε). The basic idea is that you need to look at the behavior of the sequence at the endpoints and in the middle. You then fill in a formal proof.

The following video may be helpful. It shows a proof for fn0 = xn.

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[1] Interactive Real Analysis: Sequences of Functions. Retrieved April 3, 2021 from:
[2] Convergence of Sequences of Functions: Some Additional Notes. Retrieved April 3, 2021 from:

Stephanie Glen. "Sequence of Functions, Limit Function" From Calculus for the rest of us!

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