# 2.1 Sequences and limits

A definition of a sequence of numbers is followed by the concept of convergence and limits.

A sequence of real or complex numbers is defined to be a function from the set $N$ of natural numbers into the set $R$ or $C.$ Instead of referring to such a function as an assignment $n\to f\left(n\right),$ we ordinarily use the notation $\left\{{a}_{n}\right\},$ ${\left\{{a}_{n}\right\}}_{1}^{\infty },$ or $\left\{{a}_{1},{a}_{2},{a}_{3},...\right\}.$ Here, of course, ${a}_{n}$ denotes the number $f\left(n\right).$

REMARK We expand this definition slightly on occasion to make some of our notation more indicative.That is, we sometimes index the terms of a sequence beginning with an integer other than 1. For example, we write ${\left\{{a}_{n}\right\}}_{0}^{\infty },$ $\left\{{a}_{0},{a}_{1},...\right\},$ or even ${\left\{{a}_{n}\right\}}_{-3}^{\infty }.$

We give next what is the most significant definition in the whole of mathematical analysis, i.e., what it means for a sequence to converge or to have a limit.

Let $\left\{{a}_{n}\right\}$ be a sequence of real numbers and let $L$ be a real number. The sequence $\left\{{a}_{n}\right\}$ is said to converge to $L,$ or that $L$ is the limit of $\left\{{a}_{n}\right\}$ , if the following condition is satisfied.For every positive number $ϵ,$ there exists a natural number $N$ such that if $n\ge N,$ then $|{a}_{n}-L|<ϵ.$

In symbols, we say $L=lim{a}_{n}$ or

$L=\underset{n\to \infty }{lim}{a}_{n}.$

We also may write ${a}_{n}↦L.$

If a sequence $\left\{{a}_{n}\right\}$ of real or complex numbers converges to a number $L,$ we say that the sequence $\left\{{a}_{n}\right\}$ is convergent .

We say that a sequence $\left\{{a}_{n}\right\}$ of real numbers diverges to $+\infty$ if for every positive number $M,$ there exists a natural number $N$ such that if $n\ge N,$ then ${a}_{n}\ge M.$ Note that we do not say that such a sequence is convergent.

Similarly, we say that a sequence $\left\{{a}_{n}\right\}$ of real numbers diverges to $-\infty$ if for every real number $M,$ there existsa natural number $N$ such that if $n\ge N,$ then ${a}_{n}\le M.$

The definition of convergence for a sequence $\left\{{z}_{n}\right\}$ of complex numbers is exactly the same as for a sequence of real numbers.Thus, let $\left\{{z}_{n}\right\}$ be a sequence of complex numbers and let $L$ be a complex number. The sequence $\left\{{z}_{n}\right\}$ is said to converge to $L,$ or that $L$ is the limit of $\left\{{z}_{n}\right\},$ if the following condition is satisfied.For every positive number $ϵ,$ there exists a natural number $N$ such that if $n\ge N,$ then $|{z}_{n}-L|<ϵ.$

REMARKS The natural number $N$ of the preceding definition surely depends on the positive number $ϵ.$ If ${ϵ}^{\text{'}}$ is a smaller positive number than $ϵ,$ then the corresponding ${N}^{\text{'}}$ very likely will need to be larger than $N.$ Sometimes we will indicate this dependence by writing $N\left(ϵ\right)$ instead of simply $N.$ It is always wise to remember that $N$ depends on $ϵ.$ On the other hand, the $N$ or $N\left(ϵ\right)$ in this definition is not unique. It should be clear that if a natural number $N$ satisfies this definition, then any larger natural number $M$ will also satisfy the definition. So, in fact, if there exists one natural number that works, then there exist infinitely many such natural numbers.

It is clear, too, from the definition that whether or not a sequence is convergent only depends on the “tail” of the sequence.Specifically, for any positive integer $K,$ the numbers ${a}_{1},{a}_{2},...,{a}_{K}$ can take on any value whatsoever without affecting the convergence of the entire sequence.We are only concerned with ${a}_{n}$ 's for $n\ge N,$ and as soon as $N$ is chosen to be greater than $K,$ the first part of the sequence is irrelevant.

The definition of convergence is given as a fairly complicated sentence, and there are several other ways of saying the same thing. Here are two:For every $ϵ>0,$ there exists a $N$ such that, whenever $n\ge N,$ $|{a}_{n}-L|<ϵ.$ And, given an $ϵ>0,$ there exists a $N$ such that $|{a}_{n}-L|<ϵ$ for all $n$ for which $n\ge N.$ It's a good idea to think about these two sentences and convince yourself that they really do “mean” the same thing as the one defining convergence.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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