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Posers and Puzzles

Posers and Puzzles

  1. Standard member wolfgang59
    Infidel
    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 member Palynka
    Upward Spiral
    06 Oct '10 20:43
    FAIL
  3. Standard member Palynka
    Upward Spiral
    06 Oct '10 20:44
    Repeat after me "one-to-one correspondence"... "one-to-one correspondence"... "one-to-one correspondence"...
  4. Standard member wolfgang59
    Infidel
    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 member Palynka
    Upward Spiral
    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 member wolfgang59
    Infidel
    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. 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. 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 member wolfgang59
    Infidel
    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 member Palynka
    Upward Spiral
    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. 08 Oct '10 10:37 / 1 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. 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 member wolfgang59
    Infidel
    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!