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:
M = m0/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