February 22nd, 2016, 07:02 PM  #21  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 Math Focus: Mainly analysis and algebra  Quote:
$$\sum_{n=1}^\infty a_n = \lim_{n \to \infty} \sum_{k=1}^n a_n$$ 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.  
February 22nd, 2016, 08:40 PM  #22  
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  Quote:
Quote:
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 haha 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.  
February 23rd, 2016, 01:28 AM  #23 
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  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.

February 23rd, 2016, 01:32 AM  #24  
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Quote:
 
February 23rd, 2016, 01:55 AM  #25 
Senior Member Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration 
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. 
February 23rd, 2016, 03:12 AM  #26 
Global Moderator Joined: Dec 2006 Posts: 21,018 Thanks: 2251 
...if the sum is defined as the limit of the corresponding sequence of partial sums.

February 23rd, 2016, 03:40 AM  #27  
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Quote:
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.  
February 23rd, 2016, 04:11 AM  #28 
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26 
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. 
February 23rd, 2016, 06:39 AM  #29  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 Math Focus: Mainly analysis and algebra  Quote:
From this thread, I would say that
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:
I'm glad that you apparently no longer feel that I'm flirting with crankdom though.  
February 23rd, 2016, 10:23 AM  #30  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Your wife knows that God exists through faith, not reason. Quote:
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. Last edited by Math Message Board tutor; February 23rd, 2016 at 10:46 AM.  

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