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February 22nd, 2016, 07:02 PM   #21
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Quote:
Originally Posted by Maschke View Post
The definition of the sum of an infinite series as the limit of the sequence of partial sums of the series.
This is where I'm not convinced. Not least because I don't think that's a definition in mathematics. There is a notational definition
$$\sum_{n=1}^\infty a_n = \lim_{n \to \infty} \sum_{k=1}^n a_n$$
but I don't think that is actually a definition of the sum of an infinite series. It's just a shorter way to write the limit of the sequence of partial sums of the series.

Last edited by v8archie; February 22nd, 2016 at 07:05 PM.
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February 22nd, 2016, 08:40 PM   #22
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Originally Posted by v8archie View Post
The argument boils down to the semantic distinction between assigning a value to an infinite object
Yes that is exactly what's being done.

Quote:
Originally Posted by v8archie View Post
and claiming that adding up each of the terms in that object produces that value.
NOBODY is making any such claim.

Here is what I think you think is being said:

"If we add up 1/2, and 1/4, and 1/8, ..., they will add up to 1."

Now that is said INFORMALLY. But formally, we would NEVER say that because it doesn't actually make any logical sense. We don't know how to add up infinitely many numbers. After all, the field axioms only let us add finitely many numbers. This is the point you're making when you note that finite addition is a completely different thing from infinite series. We are totally agreed.

Now here is what I think is happening, and I'll go further and say that this is the official mathematical understanding of the matter.

I come down from the mountain and say as follows: "Of course it doesn't make the slightest bit of sense to add up infinitely many numbers. After all, the field operation of addition is only defined on pairs of numbers; from which, by induction, we can prove things about addition for any finite collection of numbers. But adding infinitely many numbers makes no sense at all.

Howsomever! If we consider the sequence of partial sums 1/2, 1/2 + 1/4, etc. we can define -- I say, COMPLETELY ARBITRARILY define -- the "sum" funny ha-ha quotation marks, of the infinite series, to be the limit of the sequence of partial sums. Which is perfectly well defined once we have a rigorous construction of the real numbers along with the usual definition for the limit of a sequence.

So you are just reading way too much into this. Nobody is saying that we are adding up infinitely many things, or that this is the same addition as the + operation given by the field axioms. It isn't!! You are totally right about that.

We are just agreeing to CALL the limit of the sequence of partial sums the "sum of the infinite series" just in case that sequence of partial sums happens to have a limit.

It's just a definition, and it is NOT the same as the field addition. It's a subtle point. Yes there is a substitution going on: from the intuitive idea of an "infinite sum" to the formal definition in terms of the partial sums. These are not, strictly speaking, exactly the same thing. One is an intuition and the other is a symbolic rule. And isn't all of math full of that dichotomy!


Still ... having said all that ... you can see that it's a pretty sensible definition, right? It's a perfect clever idea, to define the sum of an infinite series as the limit of the sequence of partial sums. This clever idea lets us FINESSE all of the issues you brought up -- because you are totally correct about these issues!

But we FINESSE them with this other clever idea, and math goes on.

You see, we know that if we claimed we were adding up infinitely many things we'd be lying, because the field addition only gives us finite sums. Instead, we FINESSE all those problems. We cleverly tunnel around them. We say, Hey, we have no idea what an infinite series is. So let's just define it this other way that DOES make sense. And look, it makes everything work out! We can use this definition to prove things about infinite series. So from now on we all agree to CALL an infinite sum convergent just in case the associated sequence of partial sums has a limit. We are not making any claims about adding anything up, that's just the casual intuition.

Does that help at all? It's the truth, it's the way this is properly conceptualized if we are being pedantic. We have a logically rigorous way to define the sum of an infinite series that we can use to prove things. And people are free to have whatever private intuitions about it that they like. But no claims are made about adding infinitely many things. We're just defining what that means as something else that DOES make logical sense.

I think this is what you're saying. I agree with your philosophical points but I think you are putting math on too high a pedestal. We may have lofty ideas but in the end we have to work purely with the symbology. That is the eternal duality of math -- the interplay between the intuition and the symbology.

Last edited by Maschke; February 22nd, 2016 at 09:08 PM.
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February 23rd, 2016, 01:28 AM   #23
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Quote:
Originally Posted by greg1313 View Post
$\displaystyle \dfrac12\lim_{n\to\infty}\dfrac{1-\left(\dfrac12\right)^n}{1-\dfrac12}=1$, so the series is assigned a value of $1$.
I like this way of putting it. I see where v8Archie is coming from also. My feeling is that it makes good mathematical sense to treat the sum of the infinite series as 1. But there is also a reason for saying that this the sum will never actually reach 1 no matter how long it goes on.
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February 23rd, 2016, 01:32 AM   #24
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Quote:
Originally Posted by Maschke View Post
We are just agreeing to CALL the limit of the sequence of partial sums the "sum of the infinite series" just in case that sequence of partial sums happens to have a limit.

It's just a definition, and it is NOT the same as the field addition. It's a subtle point. Yes there is a substitution going on: from the intuitive idea of an "infinite sum" to the formal definition in terms of the partial sums. These are not, strictly speaking, exactly the same thing. One is an intuition and the other is a symbolic rule. And isn't all of math full of that dichotomy![/B]
This also is very helpful to my understanding, thanks to all for taking the question seriously.
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February 23rd, 2016, 01:55 AM   #25
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You don't understand the concept of infinity. 1+1/2+1/4...=2 It is easy to prove.

Last edited by skipjack; February 23rd, 2016 at 02:54 AM.
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February 23rd, 2016, 03:12 AM   #26
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...if the sum is defined as the limit of the corresponding sequence of partial sums.
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February 23rd, 2016, 03:40 AM   #27
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Quote:
Originally Posted by skipjack View Post
...if the sum is defined as the limit of the corresponding sequence of partial sums.
I think this what was I was getting at all along, the idea that the infinite series sums to 1 because that is the definition of how you sum an infinite series.

I'll leave the original text as it is - saying that 'the sum of the infinite series is defined as 1, as this is the limit it converges to' or words to that effect is way too pedantic.
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February 23rd, 2016, 04:11 AM   #28
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Incidentally the problem arose because of a piece of writing about Zeno's paradox of Achilles and the Tortoise. Since their respective running speeds are 5 m/s and 1 m/s, if the tortoise has a 5m headstart, Achilles will, in reality, catch up after 1.25 seconds.

However because the time period described in the paradox is 1 + 1/5 + 1/25 etc I think we could say:

Zeno is right that Achilles doesn't catch up with the tortoise if we only consider the finite 1.25 second period of time he has directed our attention towards.

Or we could say he does catch up with the tortoise after an infinite number of these increasingly small periods of time, but he still doesn't overtake him during them.

The first sentence focuses on the sum of the finite series while the latter focuses on the sum of infinite series.
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February 23rd, 2016, 06:39 AM   #29
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Quote:
Originally Posted by Maschke View Post
Here is what I think you think is being said:

"If we add up 1/2, and 1/4, and 1/8, ..., they will add up to 1."

Now that is said INFORMALLY. But formally, we would NEVER say that because it doesn't actually make any logical sense.
I rather think that many people either confuse the formal and informal phraseology or actually believe the informal phrasing.

From this thread, I would say that
  • Hedge was troubled by precisely this distinction (or rather, the lack of it); and
  • MMBt, manus and possibly complicatemodulus appear to believe the informal statement.
Only you, greg1313 and skipjack and I seem comfortable with the distinction. But it is my belief that you hold the definition too strongly. I think that since there are other methods to generate the same value for the infinite series, and because the limit expression doesn't generate values for infinite series that other methods can, that it is not correct to say that the value assigned to an infinite series is defined as the limit expression (where it exists).

Rather, I feel that the correct number to assign to the infinite limit exists independently of the limit and that, under certain requirements for convergence of the partial sums, the limit expression is one method of determining that value.
Quote:
Originally Posted by Maschke View Post
I agree with your philosophical points but I think you are putting math on too high a pedestal. We may have lofty ideas but in the end we have to work purely with the symbology. That is the eternal duality of math -- the interplay between the intuition and the symbology.
I'm not really putting the math on a pedestal other than trying to find a consistent way of approaching the theory that takes account of advances more recent than Weierstrass definition of limits. For me, the key step is to decouple the value assigned to an infinite sum from the limit calculation that we can use to determine its value.

I'm glad that you apparently no longer feel that I'm flirting with crank-dom though.
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February 23rd, 2016, 10:23 AM   #30
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Quote:
Originally Posted by v8archie View Post
Yes, my wife "knows" that God exists too.
Your wife knows that God exists through faith, not reason.

Quote:
Ah, well I'm backing up my statements with reasoned mathematical thought as opposed to dogmatic denial. We'll see where we get to.
No, you're not backing up [read: all of] your statements with reasoned mathematical thought.

Quote:
To claim that the infinite sum $\frac12 + \frac14 + \frac18 + \ldots$ actually exists and has a value is equivalent to claiming that $2 \ + \ 4 \ + \ 8 \ + \ ... \ = \ -2$ ...
This is absurd, and it can't be backed up.


Quote:
Originally Posted by Math Message board tutor

And now you've achieved forum crank status.
Quote:
Was that intended for me? How so? asked by Maschke
No, it was intended for v8archie, because you had just been referencing that similar idea.

.

Last edited by Math Message Board tutor; February 23rd, 2016 at 10:46 AM.
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