Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

VARIABLES AND LIMITS 245
range is identical with the range of x, and so you see that
in this example x and ni are, as variables, identical, two
2
symbols playing the same rôle.
(6) Consider the infinite (unending) series,
I +- + n;-I
2 22 2" 2
its terms are those of the row in (5); it is, you notice, a
geometric series or progression whose ratio, as it is called,the ratio of any term (except the first) to its predecessor,is ~; let us denote the sum of the first n terms by S.;
then, as you learned in elementary algebra, Sn=2- 2-_.
Observe that we are here confronted with three related
variables: n, n-ï, and Sn, the second being a function
of the first, and the third a function of the second (directly)
and of the first (indirectly, through the second). What
are their respective ranges? That of the first is the class
of positive integers; that of the second is the class of
Il I
numbers in the row I, - 2, -...; that of the third is
2 22) 2
3 7 I5
the class of numbers in the row I, 2, -,... I leave
2 4 16
it to you to describe the situation in the dynamic, or
picturesque, language of change, variation, behavior.
(7) The foregoing six examples are very specific. Let
us take one that is somewhat less so. Consider the
geometric progression
(S) a+ar2+ar+ +..+arF- ++..;