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Proof Without Words: Two Dimensional Geometric Series

5 Comments

  1. Graeme McRae says:

    Now, let’s see a similar 3-dimensional proof that 1/8 + 3/16 + 6/32 + 10/64 + 15/128 + … = 1

  2. Nat Silver says:

    JIm Loy sums the series on his website, using partitions:
    http://www.jimloy.com/algebra/aseries.htm A standard approach would be
    S = (1/4)(1+2x+3x^2+4x^3+…) where x = 1/2.
    S = (1/4)(d/dx(x+x^2+x^3+…))
    = (1/4)(d/dx(x/(1-x)))
    = (1/4)(1/(1-x)^2) where x = 1/2,
    as I’m sure you are aware.

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