Amazing Mathematical Sum

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Don Wells
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Amazing Mathematical Sum

Post by Don Wells »

I love this.

What is the sum of all natural numbers? That is 1 + 2 + 3 + 4 + 5 + 6 + …
Spoiler
minus 1/12 See proof here.
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Don

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HansV
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Re: Amazing Mathematical Sum

Post by HansV »

That's why mathematicians have trouble taking physicists seriously... :grin:
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Hans

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StuartR
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Re: Amazing Mathematical Sum

Post by StuartR »

That's ridiculous. I'm not convinced by the assertion that 1-1+1-1+1... = 1/2, I would have said that series does not tend towards a limit as the number of terms increases, so you can't give it a value.
StuartR


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HansV
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Re: Amazing Mathematical Sum

Post by HansV »

Stuart, that's the way mathematicians reason.

The second series is a nice example:

S2 = 1 - 2 + 3 - 4 + ...

In the video, this series is added to itself but with a shift, ending up with ¼ as the "sum".
But if you apply the same reasoning as to S1, one could claim

1
1 - 2 = -1
Average = 0
1 - 2 + 3 = 2
1 - 2 + 3 - 4 = -2
Average = 0
1 - 2 + 3 - 4 + 5 = 3
1 - 2 + 3 - 4 + 5 -6 = -3
Average = 0
So S2 = 0 :hairout:
As you say, this method is ridiculous.

Physicists, however, really do use methods like this...
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Hans

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Rudi
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Re: Amazing Mathematical Sum

Post by Rudi »

What is amazing is how enthusiastic a bloke can get by adding and subtracting 1's and 2's. :grin:
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Rudi

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HansV
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Re: Amazing Mathematical Sum

Post by HansV »

Ah, the wonders of arithmetic...
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Hans

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HansV
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Re: Amazing Mathematical Sum

Post by HansV »

By the way, using the same method, I can prove that 0 = 1:
S306.jpg
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Hans

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Rudi
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Re: Amazing Mathematical Sum

Post by Rudi »

:drop:
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Don Wells
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Re: Amazing Mathematical Sum

Post by Don Wells »

StuartR wrote:That's ridiculous. I'm not convinced by the assertion that 1-1+1-1+1... = 1/2, I would have said that series does not tend towards a limit as the number of terms increases, so you can't give it a value.
I also had trouble accepting that 1 + 1 – 1 + 1 – 1 + 1 … = ½ until I accepted infinity as a concept, not a number. The series must be allowed to go on forever.

Try this:
S1 = 1 + 1 – 1 + 1 – 1 + 1 …
1 - S1 = 1 – (1 + 1 – 1 + 1 – 1 + 1 … )
1 - S1 = 1 – 1 + 1 – 1 + 1 – 1 + 1 -1 … )
1 - S1 = S1
S1 = ½
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Don

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Don Wells
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Re: Amazing Mathematical Sum

Post by Don Wells »

HansV wrote:By the way, using the same method, I can prove that 0 = 1:
Oh Hans; you do seem to have dropped the cat amongst the pigeons. I cannot find a flaw in your argument, though I am still looking.
     :bananas:
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Don

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StuartR
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Re: Amazing Mathematical Sum

Post by StuartR »

Don,

Your argument about the value of S1 presupposes that it has a value. I don't accept that it does, it is an infinite series that does not tend towards a limit.
StuartR


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Don Wells
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Re: Amazing Mathematical Sum

Post by Don Wells »

What do we get if we sum all the natural numbers? To paraphrase Dr. Tony Padilla in an article found here on his blog, "we upset people".
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Don

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AlanMiller
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Re: Amazing Mathematical Sum

Post by AlanMiller »

HansV wrote:That's why mathematicians have trouble taking physicists seriously... :grin:
And oddly enough, many physicists don't attach much weight to string theory because it's "only" a mathematical theory... not a scientific theory. :evilgrin:

Alan