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 February 22nd, 2016, 03:39 AM #1 Senior Member   Joined: Mar 2012 Posts: 572 Thanks: 26 Approaching a limit, quick query Hi, I'm just editing some text and am uncomfortable about this sentence ½ + ¼ + 1/8 + 1/16... In mathematics this is known as an infinite series, which, despite going on forever, adds up to 1 Is there a more rigorous way of phrasing that sentence? Maybe In mathematics this is known as an infinite series, which, despite going on forever, effectively adds up to 1 ... or something like that? Any thoughts welcome.
 February 22nd, 2016, 04:39 AM #2 Senior Member   Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration What is the difference?
February 22nd, 2016, 05:38 AM   #3
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Quote:
 Originally Posted by manus What is the difference?
In the latter I'm trying to acknowledge that the infinite series is a concept but what you actually have is a finite series that approaches ever closer to summing to 1. But maybe it's OK to phrase it the first way?

February 22nd, 2016, 05:56 AM   #4
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 Originally Posted by Hedge In the latter I'm trying to acknowledge that the infinite series is a concept but what you actually have is a finite series that approaches ever closer to summing to 1. But maybe it's OK to phrase it the first way?
What do you mean by saying you actually have a finite series? The series continues for ever and ever..

February 22nd, 2016, 06:00 AM   #5
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 Originally Posted by manus What do you mean by saying you actually have a finite series? The series continues for ever and ever..

Simpler question. Is it correct to say that

A) A finite version of the series ½ + ¼ + 1/8 + 1/16... approaches a limit of 1

whereas

B) The infinite series ½ + ¼ + 1/8 + 1/16... sums to 1

February 22nd, 2016, 06:20 AM   #6
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 Originally Posted by Hedge [I] am uncomfortable about this sentence ½ + ¼ + 1/8 + 1/16... In mathematics this is known as an infinite series, which, despite going on forever, adds up to 1.
I love that you are uncomfortable about it. Through recent study of divergent series (such as the famous $\sum \limits_{n=1}^\infty n = -\frac1{12}$) I have realised more clearly that 1 is not the sum of the series you give, it is just the value we assign to the series.

However, in the case of convergent series, the value we assign definitely appears more natural than in the divergent case. We can think of it as the limit of the finite partial sums and intuitively, but not rigorously can be seen as the actual value of the infinite sum if we were able to "complete" it.

It's probably a very weak finitist viewpoint. I have no problem with the infinite set of the terms of the sum existing, but I'm less convinced that it makes sense to claim that we are actually adding them up. It's just a shorthand for saying that we can see where the sequence of sums is heading. It is similar to the idea that $$\lim_{x \to 0} f(x) = 1 \qquad \text{where} \quad f(x) = {\sin x \over x}$$ even though $f(0)$ does not exist. We can see where the function is going, and sometimes it is convenient to write $$g(x) = \begin{cases} {\sin x \over x} & x \ne 0 \\ 1 & x=0 \end{cases}$$
(occasionally we just assume it) and we can feel justified in claiming that this is the "correct" extension of the function. Essentially, we have no reason to expect that reaching the infinite case would do anything particularly strange to the function or the series, but this is just a guess on our part.

February 22nd, 2016, 06:21 AM   #7
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Quote:
 Originally Posted by Hedge Hi, I'm just editing some text and am uncomfortable about this sentence ½ + ¼ + 1/8 + 1/16... In mathematics this is known as an infinite series, which, despite going on forever, adds up to 1 Is there a more rigorous way of phrasing that sentence? Maybe In mathematics this is known as an infinite series, which, despite going on forever, effectively adds up to 1 ... or something like that? Any thoughts welcome.
How about letting $\displaystyle S_n = \sum_{k=1}^n \dfrac{1}{2^k}$, then you can say $\displaystyle \lim_{n \to \infty} S_n = 1$

February 22nd, 2016, 07:42 AM   #8
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 Originally Posted by Hedge [I]½ + ¼ + 1/8 + 1/16... In mathematics this is known as an infinite series, which, despite going on forever, adds up to 1
Quote:
 Originally Posted by v8archie I have realised more clearly that 1 is not the sum of the series you give, it is just the value we assign to the series
Then you have realized wrong.

That series does sum to 1.

.

Last edited by Math Message Board tutor; February 22nd, 2016 at 07:45 AM.

February 22nd, 2016, 08:13 AM   #9
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 Originally Posted by Math Message Board tutor Then you have realized wrong.
Well, of course, that is what you believe. But neither the mathematics nor the philosophy gives us any proof of that. $\frac12 + \frac14 + \frac18 + \ldots$ is just a way of representing $\sum \limits_{n=1}^\infty \frac1{2^n}$ which is, in turn, just a notational shorthand for $\lim_{n \to \infty} \sum \limits_{k=1}^n \frac1{2^n}$. And this last says nothing at all about truly infinite sums. It is just where the finite sums appear to be heading.

To claim that the infinite sum $\frac12 + \frac14 + \frac18 + \ldots$ actually exists and has a value is equivalent to claiming that $2 + 4 + 8 + \ldots = -2$, which is certainly a claim that makes me feel uncomfortable.

So, no. I haven't "realized wrong", I simply prefer an interpretation which is more consistent with modern mathematics and physics, than an archaic one that isn't.

February 22nd, 2016, 08:40 AM   #10
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 Originally Posted by v8archie So, no. I haven't "realized wrong", I simply prefer an interpretation which is more consistent with modern mathematics and physics, than an archaic one that isn't.
I understand the philosophical point you're making; but I do not understand your mathematical point.

Questions to clarify my understanding:

* Do you consider that pi = 3 + 1/10 + 4/100 + ...? Or do you consider only that the sequence of partial sums "approaches but never reaches" pi? In which case, do you (a) believe in the existence of the real number pi but disbelieve the existence of decimal representations of real numbers; or (b) disbelieve in the existence of pi?

* Same questions for 1/3 = .333...

* Do you believe in the least upper bound property of the real numbers? [Once you grant me this, your own theory will collapse. Once you deny this, your mathematics will collapse. Have you thought this through?]

* What do you say to the following argument? 1 + 1 isn't "really" 2. It's just a formal game. I assume (without any justification) the Peano axioms. I assume (again without justification) that 0 exists. I define 1 = S0 and 2 = SS0 and then I "prove" using the axioms that 1 + 1 = 2. But that's only a formal math game. It's not "really" true given my knowledge of modern physics. Anyone who thinks that 1 + 1 = 2 is using archaic math and physics. In fact if we smash 1 particle together with 1 particle we'll end up with slightly less than 2 particles worth of mass, since some mass will be lost as energy. So 1 + 1 = 2 is false.

* Do you feel that you are making a mathematical point? Or a philosophical one?

* If you are staking out some sort of finitist position, which sort? Do you accept 1, 2, 3, 4, ... but deny the set of natural numbers? Or do you feel that past some point, even large finite numbers such as $\displaystyle 2^{2^{2^{2^2}}}$ cease to be meaningful hence lack mathematical existence?

Last edited by Maschke; February 22nd, 2016 at 08:54 AM.

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