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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 f ( n ) , we ordinarily use the notation { a n } , { a n } 1 , or { a 1 , a 2 , a 3 , ... } . Here, of course, a n denotes the number f ( n ) .

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 { a n } 0 , { a 0 , a 1 , ... } , or even { a n } - 3 .

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

In symbols, we say L = lim a n or

L = lim n a n .

We also may write a n L .

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

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

Similarly, we say that a sequence { a n } of real numbers diverges to - if for every real number M , there existsa natural number N such that if n N , then a n M .

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

REMARKS The natural number N of the preceding definition surely depends on the positive number ϵ . If ϵ ' is a smaller positive number than ϵ , then the corresponding N ' very likely will need to be larger than N . Sometimes we will indicate this dependence by writing N ( ϵ ) instead of simply N . It is always wise to remember that N depends on ϵ . On the other hand, the N or N ( ϵ ) 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 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 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 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.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
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SUYASH Reply
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SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
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Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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Damian Reply
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Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Uday
what is nanomaterials​ and their applications of sensors.
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what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
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Damian
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Azam
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Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
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Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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