Wednesday, March 01, 2006

If we look at the formal definition for the sum of a geometric series (which I do not contest, is in mathematics a finite number) you can see that the sum of the series as n-> infinity *converges* on a finite number given by the gemoetric summation formula.

http://mathworld.wolfram.com/GeometricSeries.html

This means that the series approaches an asymptotic bound. It is a limit, it will not actually reach it. In theory, the sum of the series WILL be exactly equal to a finite number, but this will happen if and only if every one of the infinite number of elements is summed. Untill this happens, the sum is only approaching the finite number, it will still fall a bit short of it. By definition, an infinite number of elements is never ending - it is impossible to finish the task of adding all of them.

The finite value of the summation is therefore a theoretical bound, and a limit the function is approaching - it is not an actual value.

Also, lim (x-> infinity) 1/x = 0, does not require 1/infinity to equal zero - it just means that 1/infinity approaches zero and is so close, in fact, that the distance becomes negligable.

Graph 1/x - it will never actually reach 0, because y=0 is an asymptotic lower bound on the function.

Despite the fact that 1/infinity = 0 is used in math, it is still a limit, not an actual equality. There's a better explanation of this here though: http://mathforum.org/library/drmath/view/62486.html

Any thoughts?

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