Turtles all the way down:
Convergence of the Center of Mass

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
This page at http://www.pitt.edu/~jdnorton/Goodies

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The center of mass is defined as

         X = (m0x0 + m1x1 + m2x2 + ... )/M

where the total mass M is

           M = m0 + m1 + m2 + ...

Assuming that the summation forming M converges is not enough to assure that the summation defining X also converges.

For example, for M to converge, we may choose a strictly decreasing series of values for m0, m1, m2,  ... We might now assign the strictly increasing values

           x0 = 1/m0, x1 = 1/m1, x2 = 1/m2,...

We then have a value for M:

          Mm0/m0 + m1/m1 + m2/m2 + ... = 1 + 1 + 1 +  ... = ∞

It easy to arrive at cases of convergence however. Take a natural case:

         m0 = 1,  m1 = 1,  m2 =1/2,  m3 = 1/4 ...

         x0 = 0,  x1 = 1,  x2 =2,  x3 = 3 ...

Then we have

         M = 2

and

       X = 1 + 2/2 + 3/4 + 4/8 + 5/16 + 6/32 + ...

It is easy to see that this summation for X converges by expanding it as

      X = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
               +  1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
                         + 1/4 + 1/8 + 1/16 + 1/32 + ...
                                 + 1/8 + 1/16 + 1/32 + ...
                                          + 1/16 + 1/32 + ...
                                                     + ...

Summing each line individually, we recover

       X = 2 + 1 + 1/2 + 1/4 + 1/8 + ... = 4

John D. Norton, April 16, 2018