ABSTRACT
In our discussion of the computation of π, the area of the circle with radius 1, we constructed a sequence of polygons Pn with areas An. Each area An was smaller than π, and the sequence of areas increases with n,
A1 < A2 < A3 < · · · < π. The numbers An are also good approximations of π in the following sense. For any positive number , no matter how small, the difference |π − An| is smaller than for n sufficiently large. It seems to make sense to say that the numbers An approach π as n increases, or in the more common terminology, π is the limit of the sequence An. The idea of using infinite sequences to represent numbers is intrinsic
in the current notion that real numbers can be expressed as possibly infinite decimals. Consultation with our calculator indicates that π = 3.14159 . . . or
√ 2 = 1.414 . . . . These decimal representations re-
quire an infinite list of digits to achieve actual equality. The decimal expansion for π indicates that π is within 10−2 of 3.14, and is no more than 10−4 from 3.1415, etc. A similar situation arises in the summation of the geometric series.
For any number x = 1,
