2.1 Avoiding Infinite Summations (continued)

Archimedes' evaluation of K

If K were larger than 4/3, then we could inscribe triangles until their total area was more than 4/3. This would contradict equation (2.1.1) which says that the sum of the areas of the inscribed triangles is always strictly less than 4/3. If K were smaller than 4/3, then we could find an n for which is larger than K. But then equation (2.1.1) tells us that the sum of the areas of the corresponding inscribed triangles is strictly larger than K. This contradicts the fact that the sum of the areas of inscribed triangles cannot exceed the total area. Since K canot be more than 4/3 nor less than 4/3, it must equal 4/3.

This simple idea is at the heart of the modern definition of an infinite series. Archimedes was working with finite sums that approximated the value that he wanted. To a modern mathematician, an infinite series is defined by this succession of approximations by finite sums. Our finite sums may not close in quite as nicely as the geometric series, but if we can find a target value T so that for each x > T, the finite sums eventually will all be below x, and for all real numbers x < T, the finite sums eventually will all be above x, then we are allowed to say that T is the value of the infinite series. We will call this the Archimedean understanding.

Definition: Archimedean understanding The Archimedean understanding of an infinite series is that it is shorthand for the sequence of finite summations. The value of an infinite series, if it exists, is that number T such that for each x > T, the finite sums eventually will all be below x, and for each x < T, the finite sums eventually will all be above x.

For example, the Archimedean understanding of is that it is the sequence . The value of this series is 4/3 because all of the partial sums are less than 4/3, and there is no number below 4/3 that is greater than or equal to all the partial sums.

Archimedes' method of calculating areas by summing inscribed triangles is often referred to as the “method of exhaustion.” E. J. Dijksterhuis, the great Dutch historian of science, has pointed out that this is “the worst name that could have been devised,” because you never exhaust the area. You only get ever closer to it.

It may seem that Archimedes did a lot of unnecessary work simply to avoid infinite summations, but there is good reason to avoid infinite summations for they are manifestly not summations in the usual sense.

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