Converge/Diverge

freddogtgj · 2007-02-21 09:20:52

Could someone quickly explain what converge and diverge are please?

Thanks

mathsyperson · 2007-02-21 10:50:07

Those terms are usually used when describing sequences.
If a sequence converges, then as you follow it it keeps getting closer to a certain finite value.

An example of this is where the nth term is 1/n. The terms of this sequence are 1, 1/2, 1/3, 1/4, 1/5 and so on, and that sequence converges to 0, because the terms get closer and closer to 0 without ever actually being 0.

If a sequence diverges, then it eventually goes off to infinity. It can be + or - infinity, or sometimes even both. There are many examples of sequences that do this, such as where the nth term is n.

A few sequences do neither. If you've got a boring sequence where the nth term is a constant, then you can't say that it converges to that constant because it's already there, but it certainly doesn't diverge either.

More interesting functions that do neither are periodic ones, which repeat their terms in cycles.
A good example is where the nth term is (-1)^n, which gives a sequence of -1, 1, -1, 1, -1, 1...

Ricky · 2007-02-21 11:46:55

You may be going off of different definitions mathsyperson, but according to how I (and apparently MathWorld) learned convergence/divergence, those are not quite right.

A series is convergent if it's limit as approaches infinity exists. So your example of a constant series is in fact convergent. And divergent simply means "not convergent." That is, a limit does not exist. So there is no series that can be both not convergent and not divergent.

So for example, a sinusoidal curve is divergent, as it never approaches a limit.

mathsyperson · 2007-02-21 12:06:12

Ah, OK. You're right then. I haven't yet learnt properly (as in, having an authority figure tell me about them) about convergence and divergence, so the "definitions" that I posted above were just what I understood them to be.

For convergence, I just assumed that constant functions didn't count because convergence reminds me of asymptotes. Thanks for correcting me, anyway.

George,Y · 2007-02-21 16:30:29

Very good point of mathsyperson! Actually your instinct is only natural.

Many of us would think that the convergence means asymptoting or approaching. However the defination make |a[sub]n[/sub]-A|<∈ but it doesn't state that |a[sub]n[/sub]-A|>0. Therefore it allows the case of constant.

This defination turns into the one of the limit of a function instead of a series as well. Here the defination is |f(x)-A|<∈.

However, the defination emphasizes that 0 <|x-x[sub]o[/sub]|<Δ - why must it be larger than 0? Or why x, the independent variable, can only Approach a point but is Banned from BEING it? And why the difference?

Anyone having thought up an answer?? (I have one already, let's see whether ours are different)

George,Y · 2007-02-21 22:58:59

Uh-Oh, no one seems interested in this minor difference.

George,Y · 2007-02-22 23:49:20

Okay, here is my assumption.

As known to all, the limit is closely related to the derivative, especially historically. Since

it is wise to avoid the controversary Δx=0. Actually Berkeley had a fierce debate with Newton on what happens when Δt=0.
From then on, the endogenous variable is banned from arriving at the point of limit.

Accordingly, n in sequences is banned from being infinity, otherwise we will say a[sub]∞[/sub]=A in the defination, do we?

But I don't know from what time, S[sub]∞[/sub]=S. Mathematicians of today seem glad to embrace the concept of real infinity, without paying attention to its controversy.

Last edited by George,Y (2007-02-24 14:48:21)

Ricky · 2007-02-23 03:36:18

Mathematicians of today seem glad to embrace the concept of real infinity, without paying attention to its controversy.

Infinity is not a member of the real numbers, what are you talking about?

George,Y · 2007-02-23 13:56:16

I am just talking about some hidden facts that you are unaware of.

John E. Franklin · 2007-02-23 14:12:11

Converge might mean "get closer to something".
Diverge might mean "get farther away from something".
Usually you are referring to the "y" value, or height of the graph when the x value goes rightward forever or leftward forever.
If this is wrong, sorry.

Ricky · 2007-02-23 18:31:29

George,Y wrote:

I am just talking about some hidden facts that you are unaware of.

That mathematicians think infinity is a real number? Sorry, but that simply isn't true. Or do you mean something else when you say "real infinity"? If my interpretation is wrong, then please clarify.

George,Y · 2007-02-23 23:34:45

{0.9,0.99,0.999...} a[sub]∞[/sub]=1
or 0.9+0.09+0.009+...=1
--∞ entries--
Isn't that ironic?
Yes, yes, mathematicians don't explicitly use ∞ as a number, but they do believe it is an amount that can sometimes play a part.

George,Y · 2007-02-23 23:40:46

Classic Limit Theory only say that as the addition goes on, 0.9+0.09+0.009+...0.0n09 gets closer and closer to 1, OK?

Ricky · 2007-02-24 04:59:01

Classic Limit Theory only say that as the addition goes on, 0.9+0.09+0.009+...0.0n09 gets closer and closer to 1, OK?

And by the definition of what decimal expansion is, we consider equality to be when the limit approaches a number.

George,Y · 2007-02-24 14:37:05

Ricky wrote:

Classic Limit Theory only say that as the addition goes on, 0.9+0.09+0.009+...0.0n09 gets closer and closer to 1, OK?
And by the definition of what decimal expansion is, we consider equality to be when the limit approaches a number.

Yes, that is already different from the original reserved concept.
PS: "we" shall refer to the majority of modern mathematicians including Ricky.

Ricky · 2007-02-24 15:36:24

Yes, that is already different from the original reserved concept.

And what, pray tell, is the original concept?

George,Y · 2007-02-24 16:07:02

The limit concept of a function does not tell what happens on Δx=0
The limit concept of a series does not tell what happens on n=∞.
They both avoid defining the situation when the indigenous variable reach the limit point. They don't tell what the dependent variable, whether it being the function y or the series a[sub]n[/sub] or S[sub]n[/sub], really is when the indigenous variable reach the limit point, written after the rightarrow under the limit notation, whether it being x=x[sub]0[/sub], Δx=0, or n=∞.

Now you understand what I mean "classic", "historic", don't you?

Last edited by George,Y (2007-02-24 16:15:51)

Ricky · 2007-02-24 17:09:43

The limit concept of a function does not tell what happens on Δx=0

Right, because to tell what happens at Δx=0 would be the equivalent of telling the slope of a point. Points don't have slopes, so the limit concept of a function at Δx=0 is a meaningless question.

What would happen if it did tell what happens?

The limit concept of a series does not tell what happens on n=∞.

Because n can't be infinity.

They don't tell what the dependent variable, whether it being the function y or the series an or Sn, really is when the indigenous variable reach the limit point

Correct. All we care about when we talk about limits is being able to get arbitrarily close. We don't talk about reaching a limit point because we know that we can't. We accept that, and we move on. What is wrong with not being able to do something?

In that same sense, we can never, ever, write out the digits of pi. Nor all the digits of 1/3 in base 10. And we accept this. But, we're pretty darn smart. We came up with a way that we could represent pi, 1/3, and all other real numbers* by using digits. It may not be exactly what we wanted, but we can't have exactly what we wanted.

*Yes, there are a countable number of numbers that can be represented by digits, and there are an uncountable number of real numbers. But we may represent any real number to any accuracy we please.

George,Y · 2007-02-24 17:47:44

Ricky wrote:

Right, because to tell what happens at Δx=0 would be the equivalent of telling the slope of a point. Points don't have slopes, so the limit concept of a function at Δx=0 is a meaningless question.
The limit concept of a series does not tell what happens on n=∞.
Because n can't be infinity.
They don't tell what the dependent variable, whether it being the function y or the series an or Sn, really is when the indigenous variable reach the limit point
Correct. All we care about when we talk about limits is being able to get arbitrarily close. We don't talk about reaching a limit point because we know that we can't. We accept that, and we move on. What is wrong with not being able to do something?

What is wrong with not being able to do something?-Nothing wrong, but pretending you are able is wrong.

Because n can't be infinity.-
However n in S[sub]n[/sub] can really be, usually you deceive yourself by a S instead of explicit S[sub]∞[/sub]

a[sub]n[/sub]=9/10[sup]n[/sup]
S[sub]n[/sub]=1-1/10[sup]n[/sup]
S[sub]n[/sub]->1 when n->∞
or

in standard form

Therefore S (=S[sub]∞[/sub]) = 1?????????

Yes you write carefully without any ∞ appearing, but the sum of the "whole" of 0.9+0.09+0.009... does tell the essence of infinite sum- from knowing the approaching, define the reaching.

Besides, Δx≠0 is not that the mathematicians didn't allow the slope of a point. Rather, it was because they didn't know the situation of Δy/Δx or Δs/Δt when the denominator is zero, empty.

They didn't know, so they left it blank.

How about you, Ricky? You don't know so you define it?

Last edited by George,Y (2007-02-24 17:49:58)

Ricky · 2007-02-24 20:11:39

Yes you write carefully without any ∞ appearing, but the sum of the "whole" of 0.9+0.09+0.009... does tell the essence of infinite sum- from knowing the approaching, define the reaching.

When we talk about infinite sums, we talk about them in terms of limits simply because there is no other way to talk about them. Because we always talk about them in terms of limits, and everyone who deals with them knows we are talking about limits, we simply leave the actual limit notation out to make it easier to write.

When we talk about infinite decimal expansion, we again are talking about limits. Just because no one explicitly says "limit" in either of these cases does not mean that limit is not implied.

George,Y · 2007-02-24 20:49:51

Ricky wrote:

we can find a delta which suffices. If you define epsilon to be 100,000,000,000,000^-100,000,000,000,000,000,000, again, we can find a delta. We can get as abitrarily close to 1 as we want.
Now go back to real analysis. Write out each s_n, s_n being the sum of all terms up to n.
s_0 = 0
s_1 = 0.9
s_2 = 0.99
s_3 = 0.999
s_4 = 0.9999
And so on. We define the summation of an infinite series to be the limit of this series as it approaches infinity.
If you don't accept that definition, that's fine. You don't have to. However, by not accepting it, you throw most, if not all, of real analysis out the window.

from this page

When we talk about infinite decimal expansion, we again are talking about limits.
I don't know whom the "we" includes, please ask Dross and Zhylliolum(sorry if misspelled) about the meaning of infinite decimal.

Besides, what does the limit of (1+1/n)[sup]n[/sup] approaches? It approaches the limit of
2+0.7+0.01+0.008+0.0002+0.00008+0.000001+0.0000008+0.00000002+0.000000008+...
That's fairly interesting.

George,Y · 2007-02-24 23:07:52

Let's see what "we" wrote about what 0.999... is

Dross wrote:

There is a well-defined concept of what a limit is.
A human's inability to calculate a number using an infinite number of decimal places is very much not what is at issue here. Just because you could never actually add them by writing on a piece of paper does not mean anything as far as the decimal expansion is concerned.
the sum of 0.9(recuring) does not have to "reach" it's "last" digit (even though we might define it in a way that might suggest that it does, this is again our shortcoming) - it is not "in time", so to speak. Every digit has already been added, it always has been and always will be.
A human must add them one at a time, but I suppose you could think of adding them all at once. I really don't know how to express it another way - I've given you a sound argument which basically says that 0.9(recuring) = 1, simply by the way we define the number 0.9(recuring).

Post51

Ricky wrote:

What do you mean interpret? It is a 0. and then an infinite number of 9's after it.
Please point out an error in my post #76.

Post94

Ricky wrote:

This is the only criticism of my proof that I could find in your post. And an infinite number has no end. What is your problem with that?

Post111

luca-deltodesco wrote:

tbh, i dont understand how anyone can argue this simple proof? you get people saying, oh but when you times it by 10, there must be on less 9 on the end of it, but that doesnt make sense, because thats treating infinity as a definate value.

Post135

MathsIsFun wrote:

If you try to multiply 4.999... one digit at a time you won't ever get an answer, but if you step back and see the *idea* then you will find that you can.
For example
4.9 × 3 = 14.7
4.99 × 3 = 14.97
4.999 × 3 = 14.997
So, the pattern is that 4.999... = 14.999...
And what about that last 7? You will *never* reach it. Infinity is endless.

Post197

MathsIsFun wrote:

Infinite is not "big", it is endless. Nothing in our real world is like it. But it is a simple idea. It is simply "no end".

Post234

kylekatarn wrote:

It wasn't a suggestion. Someone who doesn't ""believe"" 0.(9) = 1 it's because one of two things:
- he/she doesn't really know that 0.(9) is formed by the infinite sum 0.9 + 0.09 + 0.009 + ... = 1

Post336

The quotes above clearly show people consider infinite sum as possible thus being rather than closing possilble.

The actual problem, Ricky, is not that someone denies

or that (0.9,0.99,0.999...} gets closer and closer to 1
but whether you can state 0.999...(infinite digits)=1.

You may say when you say so, you mean the same as the previous because you have defined it. Sorry that's not reasonable. Because "being" and "closing" are different in language. So are finite n digits of 0.999999 and infinite 0.9999.... The way you use a definition to equate the two is as I used a defintion to equate holding a gun as killing people. (I can also define killing people as holding a gun) The misunderstanding is inevitable.

However, you may do something for better communication. Every time you wanna prove 0.999...=1
You shall state this equation in your language is just {0.9, 0.99, 0.999, ...}gets closer and closer to 1 and nothing else. But that is an absurd explaination, isn't it? S/he may reply that what is the case of 0.999... (infinite digits). You shall frankly shrug and say" I don't know, but I'd like to define it."

Also, never force others to accept your definition, or simply put, your language.

Last edited by George,Y (2007-02-25 01:33:46)

George,Y · 2007-02-24 23:20:21

,even in the name of "proof".

Ricky · 2007-02-25 05:48:09

The actual problem, Ricky, is not that someone denies

or that (0.9,0.99,0.999...} gets closer and closer to 1
but whether you can state 0.999...(infinite digits)=1.

But, what we mean by 0.999... is that its getting closer and closer to 1 the more digits you write out. And it's what we mean when we say 0.333... Just because we don't explicitly say "limit" doesn't mean its not there.

freddogtgj · 2007-02-26 09:21:39

thanks people for the help!

Math Is Fun Forum

#1 2007-02-21 09:20:52

Converge/Diverge

#2 2007-02-21 10:50:07

Re: Converge/Diverge

#3 2007-02-21 11:46:55

Re: Converge/Diverge

#4 2007-02-21 12:06:12

Re: Converge/Diverge

#5 2007-02-21 16:30:29

Re: Converge/Diverge

#6 2007-02-21 22:58:59

Re: Converge/Diverge

#7 2007-02-22 23:49:20

Re: Converge/Diverge

#8 2007-02-23 03:36:18

Re: Converge/Diverge

#9 2007-02-23 13:56:16

Re: Converge/Diverge

#10 2007-02-23 14:12:11

Re: Converge/Diverge

#11 2007-02-23 18:31:29

Re: Converge/Diverge

#12 2007-02-23 23:34:45

Re: Converge/Diverge

#13 2007-02-23 23:40:46

Re: Converge/Diverge

#14 2007-02-24 04:59:01

Re: Converge/Diverge

#15 2007-02-24 14:37:05

Re: Converge/Diverge

#16 2007-02-24 15:36:24

Re: Converge/Diverge

#17 2007-02-24 16:07:02

Re: Converge/Diverge

#18 2007-02-24 17:09:43

Re: Converge/Diverge

#19 2007-02-24 17:47:44

Re: Converge/Diverge

#20 2007-02-24 20:11:39

Re: Converge/Diverge

#21 2007-02-24 20:49:51

Re: Converge/Diverge

#22 2007-02-24 23:07:52

Re: Converge/Diverge

#23 2007-02-24 23:20:21

Re: Converge/Diverge

#24 2007-02-25 05:48:09

Re: Converge/Diverge

#25 2007-02-26 09:21:39

Re: Converge/Diverge

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