Is this a legitimate geometric proof that the infinite sum of 1/(2^n) = 1?

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Mates

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Wikipedia has what appears to be a geometric proof (at the very top right of the page) saying that the infinite sum of 1/2^n = 1 (from n = 1) here, https://en.wikipedia.org/wiki/Geometric_series .

It doesn't say anything about it being a proof, but I am wondering if it is a legitimate proof?
 
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Mates said:
Wikipedia has what appears to be a geometric proof saying that the infinite sum of 1/2^n = 1 (from n = 1) here, https://en.wikipedia.org/wiki/Geometric_series .

It doesn't say anything about it being a proof, but I am wondering if it is a legitimate proof?
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Which picture are you referring to? There are several such visual proofs shown there, one of which shows the particular fact you mention:

330px-Geometric_Segment.svg.png

Each of them can be turned into a formal proof, though in itself it needs a good bit of explanation. What they do is to convince you of the truth of the fact, if you think enough about the picture. As Wikipedia says,

A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.​
 
Dr.Peterson said:
Which picture are you referring to? There are several such visual proofs shown there, one of which shows the particular fact you mention:

330px-Geometric_Segment.svg.png

Each of them can be turned into a formal proof, though in itself it needs a good bit of explanation. What they do is to convince you of the truth of the fact, if you think enough about the picture. As Wikipedia says,

A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.​
Click to expand...
Thanks for catching that. I was looking at the illustration at the top right of the page, but the illustration right below this one that you show in your post is another example that I am interested in.

I am not sure exactly what words would be required. I suppose I am wondering if there even are words to go along with those illustrations that are sufficient to be a formal proof.
 
Mates said:
Thanks for catching that. I was looking at the illustration at the top right of the page, but the illustration right below this one that you show in your post is another example that I am interested in.

I am not sure exactly what words would be required. I suppose I am wondering if there even are words to go along with those illustrations that are sufficient to be a formal proof.
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You are talking about the sum of an infinite series. By definition, this sum is the limit of the sequence of partial sums of that series. Therefore, the "required" words would be:
1. Formal definition of the series
2. Formal proof that the limit of its partial sums is the number in question.
 
Mates said:
I was looking at the illustration at the top right of the page, but the illustration right below this one that you show in your post is another example that I am interested in.
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Neither of those is for the specific series you referred to. You apparently misquoted them.
Mates said:
I am not sure exactly what words would be required. I suppose I am wondering if there even are words to go along with those illustrations that are sufficient to be a formal proof.
Click to expand...
Try describing what you do see, in just a little more detail than they give in the caption, and then think about what parts need still more explanation.

This will not result in a fully formal proof (which would be far more concise, as @lev888 implies), but would be a bridge between that and intuitive understanding. Again, what Wikipedia said was, "However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof." They don't say you would actually use them as the start of a proof.

The test of a visual proof is simply whether it convinces you.
 
lev888 said:
You are talking about the sum of an infinite series. By definition, this sum is the limit of the sequence of partial sums of that series. Therefore, the "required" words would be:
1. Formal definition of the series
2. Formal proof that the limit of its partial sums is the number in question.
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Yes thank you, but I know about that definition. I was hoping to know if the diagram from the link can be used to make a formal proof without using the definition.
 
Mates said:
Yes thank you, but I know about that definition. I was hoping to know if the diagram from the link can be used to make a formal proof without using the definition.
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A formal proof without the definition? How's that? How do you prove that an oak is a tree without the definition of a tree?
 
Dr.Peterson said:
Neither of those is for the specific series you referred to. You apparently misquoted them.
Click to expand...
The diagram below the one that you posted appears to show the series 1/2^k (from k =0 to infinity). I thought that the point of that diagram is to show that the sum of those areas equals the sum of 1/2^k (from k =0 to infinity), which is 2.

Try describing what you do see, in just a little more detail than they give in the caption, and then think about what parts need still more explanation.

This will not result in a fully formal proof (which would be far more concise, as @lev888 implies), but would be a bridge between that and intuitive understanding. Again, what Wikipedia said was, "However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof." They don't say you would actually use them as the start of a proof.

The test of a visual proof is simply whether it convinces you.
Click to expand...
Unfortunately for me, I have learnt that my "proofs" can be wrong even if I am absolutely sure they are correct. I was hoping to know if a proof for this exists (without the use of the definition of the sum).
 
lev888 said:
A formal proof without the definition? How's that? How do you prove that an oak is a tree without the definition of a tree?
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I only mean that I don't want to use the definition of the sum in this particular case. I know we need definitions at some point.

I am just hoping to find way to deduce (from other definitions) that an infinite sum of 1/2^n = 1 (from n = 1 to infinity). Or more specifically, I would like to know if the diagrams on the Wikipedia page can be used to make another formal proof.
 
Mates said:
The diagram below the one that you posted appears to show the series 1/2^k (from k =0 to infinity). I thought that the point of that diagram is to show that the sum of those areas equals the sum of 1/2^k (from k =0 to infinity), which is 2.
Click to expand...
Yes; but what you said was,
Mates said:
Wikipedia has what appears to be a geometric proof (at the very top right of the page) saying that the infinite sum of 1/2^n = 1 (from n = 1)
Click to expand...
That's why I couldn't be sure which one you meant.

Mates said:
I am just hoping to find way to deduce (from other definitions) that an infinite sum of 1/2^n = 1 (from n = 1 to infinity). Or more specifically, I would like to know if the diagrams on the Wikipedia page can be used to make another formal proof.
Click to expand...
I wouldn't really try to make a formal proof from that.
Mates said:
Unfortunately for me, I have learnt that my "proofs" can be wrong even if I am absolutely sure they are correct. I was hoping to know if a proof for this exists (without the use of the definition of the sum).
Click to expand...
What exactly do you mean by that?
 
Mates said:
I am interested to see if there is another way to prove that the sum of 1/2^n = 1 (from n = 1 to infinity).
Click to expand...
There are many ways.

But they all depend on defining what you are asking about. What do you mean by "without the use of the definition"?
 
Dr.Peterson said:
There are many ways.

But they all depend on defining what you are asking about. What do you mean by "without the use of the definition"?
Click to expand...
The definition I was referring to is the limit of the partial sum of the first n terms is also the sum of the series. In this case, the limit of the partial sums is 1, so by definition the sum is 1 too.

But I am more interested in learning other ways of proving the sum of 1/2^n = 1 (from n = 1 to infinity).
 
This does not prove that the limit exists, but if it does then:
[math]S = \sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2} + \sum_{n=2}^\infty \frac{1}{2^n} = \frac{1}{2} + \frac{1}{2} \sum_{n=2}^\infty \frac{1}{2^{n-1}} = \frac{1}{2} + \frac{1}{2} \sum_{k=1}^\infty \frac{1}{2^k} = \frac{1}{2} + \frac{1}{2} S[/math]
 
blamocur said:
This does not prove that the limit exists, but if it does then:
[math]S = \sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2} + \sum_{n=2}^\infty \frac{1}{2^n} = \frac{1}{2} + \frac{1}{2} \sum_{n=2}^\infty \frac{1}{2^{n-1}} = \frac{1}{2} + \frac{1}{2} \sum_{k=1}^\infty \frac{1}{2^k} = \frac{1}{2} + \frac{1}{2} S[/math]
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I think this proof assumes that the sum equals the limit. I am hoping to see a proof that doesn't require that assumption.
 
This diagram is another illustration, but I do not consider it (or most other geometric illustrations) to be a formal proof. If you need a formal proof, the simplest way is using the summation of geometric sequences to show that
[math]s_n = \sum_{k=1}^n = 1 - \left(\frac{1}{2}\right)^n[/math]then prove that the limit of [imath]s_n[/imath] is 1.
 
The area of a 1 by 1 square is 1(unit).
The picture adds up the area in a special way. Yes, this is a proof that shows that 1= 1/2+ 1/4 + 1/8 + 1/16 +...
 
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