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Posers and Puzzles

Posers and Puzzles

  1. Standard member royalchicken
    CHAOS GHOST!!!
    30 Jun '04 19:52
    Think of a clever way to show that 1 + 1/4 +1/9 + 1/16 + ... is between 3/2 and 7/4...
  2. Donation Acolyte
    Now With Added BA
    30 Jun '04 20:20 / 1 edit
    Let

    A = 1 + 1/4 + 1/9 + 1/16 + ...

    B = 1/6 + 1/12 + 1/20 + 1/30 + ...

    C = 1 + B

    D = 1 + 1/4 + B

    Then B = (1/2 -1/3) + (1/3 - 1/4) + ...
    = 1/2 + (-1/3 + 1/3) + (-1/4 + 1/4) + ... (can regroup terms like this as they tend to 0)
    = 1/2

    => C = 3/2, D = 7/4

    A lies between C and D by comparison.

    Edit: This is how I'd have done it. Did you have a cleverer way in mind?
  3. Standard member royalchicken
    CHAOS GHOST!!!
    04 Jul '04 00:55
    Originally posted by Acolyte
    Let

    A = 1 + 1/4 + 1/9 + 1/16 + ...

    B = 1/6 + 1/12 + 1/20 + 1/30 + ...

    C = 1 + B

    D = 1 + 1/4 + B

    Then B = (1/2 -1/3) + (1/3 - 1/4) + ...
    = 1/2 + (-1/3 + 1/3) + (-1/4 + 1/4) + ... (can regroup terms like this as they tend to 0)
    = 1/2

    => C = 3/2, D = 7/4

    A lies between C and D by comparison.

    Edit: This is how I'd have done it. Did you have a cleverer way in mind?
    Your way is certainly cleverer than the way the examiner wanted me to do it, which also ended up being the way I did it, funnily enough. (Your way is rather shorter anyway...)