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    Why does

     1 - 1 + 1 - 1 + 1 - 1........ = \frac{1}{2}?

    I mean how can it be a fraction in the first place?
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    (Original post by gangsta316)
    Why does

     1 - 1 + 1 - 1 + 1 - 1........ = \frac{1}{2}?

    I mean how can it be a fraction in the first place?
    It doesn't equal a half...
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    It doesn't, not in the conventional way of summing anyway.
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    What on earth make su think it does?
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    Well the only way that can be a half is using a flawed argument:

    It's easy to make is equal 0 or 1 just consider:

    (1-1)+(1-1)+(1-1)+....
    or
    1- (1-1)-(1-1)-(1-1)-...

    Now for it to equl a half consider using the sum to infinity for a geometric series:

    Sum to infinity of a G.P. = 1/(1-r)

    However this is only valid for l r l<1 thus let r take these values and see what happens as r tends to -1.

    we get Sum to infinity = 1/2

    edit: Sorry about the lack of Latex
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    (Original post by gangsta316)
    Why does

     1 - 1 + 1 - 1 + 1 - 1........ = \frac{1}{2}?

    I mean how can it be a fraction in the first place?
    I suspect gangsta has encountered this sort of dubious argument. Something similar was posted on a thread a week or so ago.

    x = 1 - 1 + 1 - 1 + 1 - 1 ....

    1 - x = 1 - 1 + 1 - 1 + 1 - 1 ....

    1 - x = x

    1 = 2x

    x = \frac{1}{2}
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    (Original post by Mr M)
    I suspect gangsta has encountered this sort of dubious argument. Something similar was posted on a thread a week or so ago.

    x = 1 - 1 + 1 - 1 + 1 - 1 ....

    1 - x = 1 - 1 + 1 - 1 + 1 - 1 ....

    1 - x = x

    1 = 2x

    x = \frac{1}{2}

    I like!

    THanks for the proof Mr M.
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    (Original post by wizz_kid)
    I like!

    THanks for the proof Mr M.
    Please note that I said it was a dubious argument wizz!

    The terms of this series do not converge to zero as the number of terms increases so it is not meaningful to find the sum.
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    (Original post by Mr M)
    Please note that I said it was a dubious argument wizz!

    The terms of this series do not converge to zero as the number of terms increases so it is not meaningful to find the sum.
    So if it is not = half then what does it sum to?
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    (Original post by wizz_kid)
    So if it is not = half then what does it sum to?
    It doesn't sum to anything. It is divergent - ie doesn't have a sum
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    (Original post by SimonM)
    It doesn't sum to anything. It is divergent - ie doesn't have a sum

    Oh ic.

    Srry, how do u know that it doesnt have a sum?
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    (Original post by wizz_kid)
    Oh ic.

    Srry, how do u know that it doesnt have a sum?
    A sequence has a sum if the "partial sums" converge

    ie. The sequence \{ a_1, a_2, \ldots , a_k, \ldots \} has a sum iff

    The sequence \displaystyle s_n = \sum_{k=1}^n a_1 converges.

    Since s_n = 1, 0, \ldots , 0, 1, 0, \ldots it doesn't ever converge
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    Think of it as a graph; it would look like some sort of infinitely wide square wave, which doesn't converge to any particular value.
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    (Original post by ashy)
    Think of it as a graph; it would look like some sort of infinitely wide square wave, which doesn't converge to any particular value.

    AAh ic!

    Physicists are smart lol!
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    (Original post by SimonM)
    A sequence has a sum if the "partial sums" converge

    ie. The sequence \{ a_1, a_2, \ldots , a_k, \ldots \} has a sum iff

    The sequence \displaystyle s_n = \sum_{k=1}^n a_1 converges.

    Since s_n = 1, 0, \ldots , 0, 1, 0, \ldots it doesn't ever converge

    Makes sense!

    THanks.
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    wizz,

    You can get into all sorts of difficulties if you try to sum divergent series.

    Consider this:

    x = 1 + 2 + 3 + 4 + 5 + 6 + ....

    As x is the sum of positive numbers, it must be positive.

    2x = 2 + 4 + 6 + 8 + 10 + 12 + ....

    x = 2x - x = -1 - 3 - 5 - 7 - 9 + ....

    As x is the sum of negative numbers, it must be negative.

    So x is simultaneously positive and negative.
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    @OP: What you're referring to is summing the series using Cesáro summation. Using this method, the sum is indeed 1/2. However, using conventional summation, the series is divergent and so does not have a sum.
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    (Original post by Mr M)

    So x is simultaneously positive and negative.

    I think i get it .

    THanks Mr M
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    No problem.
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    Cesáro summation is sort of like "taking the average" (more specifically, taking the limit of averages) so that's how you can get a fractional 1/2 out of 1's and 0's.
 
 
 
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