What is mathematics?

There is a video doing the rounds that shows that 1 + 2 + 3 + … = -1/12. For those like me who don’t like videos, here is the math. It involves nothing more than simple algebra and it is not a hoax. Apparently, Euler reached the same result.

S1 = 1 – 1 + 1 – 1 + …
1 – S1 = 1 – 1 + 1 – 1 + … = S1
S1 = 1/2
S2 = 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + …
S2 =     1 – 2 + 3 – 4 + 5 – 6 + 7 – …
2 S2 = 1 – 1 + 1 – 1 + … = S1
S2 = 1/4
S = 1 + 2 + 3 + 4 + …
S – S2 = 4 + 8 + 12 + 16 + … = 4S
S = -1/12

The reason for this highly non-intuitive result is that all of these sums are divergent series, and the equality used in the second line of the math above doesn’t hold in a strict sense.

Apparently summations like this are useful even if they seem to make no sense. Somewhat like complex numbers.

And that fact raises the question that is the title of the post. More on this later when I get some time to ponder it.

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4 Responses

  1. I have been reading your blog for a while but I haven’t ever commented. I could not help but pointing out that this is false.

    The sum of the numbers from 1 to inifinity is equal to infinity. There is not a way in which positive numbers can add to be a negative.

    The error lies in the first second statement. The first series S1 = 1 – 1 + 1 – 1 + … is equal to 0. (1-1) + (1-1) +… (1-1) = 0 The second statement is completely wrong. If S1 = 0 then 1-S1 != 0, rather 1-S1 = 1.

    Another way to show the error is this:

    Assume 1-S1 = S1.
    If S1 = 0 then 1-0 = 0 is the result which is not true.
    If S1 = 1 then 1-1 = 1 is the result which is not true.

  2. John,
    The technique used to perform the summations is certainly dubious (algebra on divergent series is not allowed), if not completely wrong, but the results are not false.

    They are limiting cases of perfectly valid convergent sequences. See Ramanujan summation (http://en.wikipedia.org/wiki/Ramanujan_summation) for more details.

    In some mathematical sense then, the results are true. So the question is, what are we doing when we use such results? What is the connection of mathematics with reality (in which there are no infinities)?

  3. waiting for your next post

  4. I wonder how a divergent series can have a result other than infinity. It kind of goes against all real analysis that I have read of.

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