Class notes

sequence and series Functions

0 purchase

Course
Statistics (SSTA031)

Institution
University Of Limpopo (UL)

A series of functions is a collection of functions indexed by a parameter, while a sequence is an ordered list of elements indexed by natural numbers.

[Show more]

Preview 2 out of 15 pages

View example

Uploaded on March 21, 2024
Number of pages 15
Written in 2023/2024
Type Class notes
Professor(s) Prof shabalala
Contains All classes

statistics

R50,00

Add to cart

Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

, Chapter 9

Sequences and Series of
Functions

In this chapter, we define and study the convergence of sequences and series of
functions. There are many different ways to define the convergence of a sequence
of functions, and different definitions lead to inequivalent types of convergence. We
consider here two basic types: pointwise and uniform convergence.

9.1. Pointwise convergence
Pointwise convergence defines the convergence of functions in terms of the conver-
gence of their values at each point of their domain.
Definition 9.1. Suppose that (fn ) is a sequence of functions fn : A → R and
f : A → R. Then fn → f pointwise on A if fn (x) → f (x) as n → ∞ for every
x ∈ A.

We say that the sequence (fn ) converges pointwise if it converges pointwise to
some function f , in which case
f (x) = lim fn (x).
n→∞

Pointwise convergence is, perhaps, the most obvious way to define the convergence
of functions, and it is one of the most important. Nevertheless, as the following
examples illustrate, it is not as well-behaved as one might initially expect.
Example 9.2. Suppose that fn : (0, 1) → R is defined by
n
fn (x) = .
nx + 1
Then, since x 6= 0,
1 1
lim fn (x) = lim = ,
n→∞ n→∞ x + 1/n x

167

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying this summary from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller napemahlwele. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy this summary for R50,00. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

68175 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy summaries for 15 years now

Start selling