Posers and Puzzles

Posers and Puzzles

  1. Standard memberwolfgang59
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    06 Oct '10 20:25
    1+2+3+4+ ... equals?

    ZERO

    Obviously 1+2+3+4+ ... = (2-1)+(3-1)+(4-1)+(5-1)+...

    =2+3+4+5+... - (1+1+1+1+...)

    = inf - inf

    = 0
  2. Standard memberPalynka
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    06 Oct '10 20:43
    FAIL
  3. Standard memberPalynka
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    06 Oct '10 20:44
    Repeat after me "one-to-one correspondence"... "one-to-one correspondence"... "one-to-one correspondence"...
  4. Standard memberwolfgang59
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    06 Oct '10 21:21
    😕

    Of course!

    I'm trying to point out the flaw in our opponents' argument in the "many balls" thread
  5. Standard memberPalynka
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    06 Oct '10 21:37
    Originally posted by wolfgang59
    😕

    Of course!

    I'm trying to point out the flaw in our opponents' argument in the "many balls" thread
    Are there more even numbers than natural numbers?

    Yet if I start with this sequence

    {2} and {1,2}
    {2,4} and {1,2,3,4}
    ...
    {2,...,2n} and {1,2,...,n,...,2n}
    ...
    n goes to infinity...
    ...

    The difference in cardinality is always increasing in n!

    Yet...the cardinality of the countably infinite sets is the same.

    Now repeat after me "one-to-one correspondence"...
  6. Standard memberwolfgang59
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    06 Oct '10 22:31
    Originally posted by Palynka
    Are there more even numbers than natural numbers?

    Yet if I start with this sequence

    {2} and {1,2}
    {2,4} and {1,2,3,4}
    ...
    {2,...,2n} and {1,2,...,n,...,2n}
    ...
    n goes to infinity...
    ...

    The difference in cardinality is always increasing in n!

    Yet...the cardinality of the countably infinite sets is the same.

    Now repeat after me "one-to-one correspondence"...
    yes but did you know that there are more evens than odds?

    for the odd nmber 1 there is 1+1=2

    for the odd nmber 3 there is 3+1=4

    for the odd nmber 3 there is 5+1=6

    for the odd nmber Nthere is N+1

    So for every odd there is an even ... but then we include ZERO

    QED There are more evens than odds







    😉
  7. Joined
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    07 Oct '10 04:28
    Originally posted by wolfgang59
    1+2+3+4+ ... equals?

    ZERO

    Obviously 1+2+3+4+ ... = (2-1)+(3-1)+(4-1)+(5-1)+...

    =2+3+4+5+... [b]-
    (1+1+1+1+...)

    = inf - inf

    = 0[/b]
    inf - inf is not zero. It's not even defined. Therefore your calculation is flawed.

    If we define inf - inf, inf / inf, and such, as a number then we get funny results, as the one above.
  8. ALG
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    07 Oct '10 06:10
    Did you know that 0 = 1?

    0 = 0 + 0 + 0 + ...
    = (1-1) + (1-1) + (1-1) + ...
    = 1 + (-1+1) + (-1+1) + (-1+1) + ....
    = 1

    QED
  9. Standard memberwolfgang59
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    07 Oct '10 12:21
    Originally posted by FabianFnas
    inf - inf is not zero. It's not even defined. Therefore your calculation is flawed.

    If we define inf - inf, inf / inf, and such, as a number then we get funny results, as the one above.
    I think I knew that!
    😕
  10. Standard memberPalynka
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    08 Oct '10 10:21
    Originally posted by Thomaster
    Did you know that 0 = 1?

    0 = 0 + 0 + 0 + ...
    = (1-1) + (1-1) + (1-1) + ...
    = 1 + (-1+1) + (-1+1) + (-1+1) + ....
    = 1

    QED
    So which standard operation breaks down when you reach an infinite sum? Could you get that result by extrapolating from the union of finite sequences of sums?
  11. Joined
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    08 Oct '10 10:371 edit
    If I remember correctly, it's all to do with whether the series is absolutely convergent or not.

    SUM(x_i) is absolutely convergent <=> SUM(|x_i|) is convergent.

    Absolutely convergent series behave when you rearrange them. Series that are just convergent (e.g. 1 - 1/2 + 1/3 - 1/4 + ...) can converge to different values (or not at all) depending on the order.

    For example:
    1 - 1/2 + 1/3 - 1/4 + 1/5 +... = ln 2
    1 + 1/3 + 1/5 + ..... - 1/2 - 1/4 - 1/6 - ... does not converge

    1 - 1 + 1 - 1 + ... is very obviously not absolutely convergent!
  12. Joined
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    08 Oct '10 20:09
    Originally posted by wolfgang59
    I think I knew that!
    😕
    I know you know. And I know also that you know that I know that you know it.
  13. Standard memberwolfgang59
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    08 Oct '10 20:27
    Originally posted by FabianFnas
    I know you know. And I know also that you know that I know that you know it.
    I didnt know that!
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