# 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.

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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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