Real Analysis – Sequences of functions
7.1 Motivation
Up to now we have explored functions of a single real variable and sequences of real numbers.
however, in many different fields ranging from economics to electronic engineering, we often
encounter sequences of functions. So, the first question that comes to your mind is what is a
sequence of functions? A basic example to this would be
𝑓𝑛(𝑥) ∶= 𝑥𝑛 .
Which, in a sense can be considered as a function of two variables 𝑥 and 𝑛. However, that is not
what we are going to do here. Instead, we can put it as a real valued function 𝑓𝑛 ; for each (fixed)
𝑛 ∈ ℕ or as a sequence of real numbers 𝑓𝑛(𝑥) ; for each (fixed) 𝑥 ∈ ℝ . Some basic questions
that we are going to answer here are:
1) Is it possible to do lim lim 𝑓𝑛(𝑥) = lim lim 𝑓𝑛(𝑥) in any situation?
𝑛→∞ 𝑥→𝑎 𝑥→𝑎 𝑛→∞
2) Can you differentiate an infinite series of functions (i.e. Taylor’s series) term by term?
Which all of you may have taken for granted when doing advanced level past questions.
𝑏 𝑏
3) What about lim ∫ 𝑓 (𝑥) 𝑑𝑥 = ∫ lim 𝑓 (𝑥) 𝑑𝑥 ?
𝑛 𝑎 𝑛→∞ 𝑛
𝑛→∞ 𝑎
As this is an introductory real analysis course we will restrict our rigorous treatment to 1) above.
Two practical applications of this theory are given in the next pages. However, solving these
applications is far beyond our scope and are suitable for senior students.
Profit problem: https://personal.lse.ac.uk/sasane/ma412.pdf
Electronic and Electrical Engineering: http://iris.elf.stuba.sk/JEEEC/data/pdf/03-04_101-08.pdf
1
, 2
7.1 Motivation
Up to now we have explored functions of a single real variable and sequences of real numbers.
however, in many different fields ranging from economics to electronic engineering, we often
encounter sequences of functions. So, the first question that comes to your mind is what is a
sequence of functions? A basic example to this would be
𝑓𝑛(𝑥) ∶= 𝑥𝑛 .
Which, in a sense can be considered as a function of two variables 𝑥 and 𝑛. However, that is not
what we are going to do here. Instead, we can put it as a real valued function 𝑓𝑛 ; for each (fixed)
𝑛 ∈ ℕ or as a sequence of real numbers 𝑓𝑛(𝑥) ; for each (fixed) 𝑥 ∈ ℝ . Some basic questions
that we are going to answer here are:
1) Is it possible to do lim lim 𝑓𝑛(𝑥) = lim lim 𝑓𝑛(𝑥) in any situation?
𝑛→∞ 𝑥→𝑎 𝑥→𝑎 𝑛→∞
2) Can you differentiate an infinite series of functions (i.e. Taylor’s series) term by term?
Which all of you may have taken for granted when doing advanced level past questions.
𝑏 𝑏
3) What about lim ∫ 𝑓 (𝑥) 𝑑𝑥 = ∫ lim 𝑓 (𝑥) 𝑑𝑥 ?
𝑛 𝑎 𝑛→∞ 𝑛
𝑛→∞ 𝑎
As this is an introductory real analysis course we will restrict our rigorous treatment to 1) above.
Two practical applications of this theory are given in the next pages. However, solving these
applications is far beyond our scope and are suitable for senior students.
Profit problem: https://personal.lse.ac.uk/sasane/ma412.pdf
Electronic and Electrical Engineering: http://iris.elf.stuba.sk/JEEEC/data/pdf/03-04_101-08.pdf
1
, 2
