2 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

the explanations that are to follow will make the meaning
clear.

4. In the first place the word * series” (or ‘ progression ’’)
as used here denotes a class of things arranged in order so
that one of its elements, or members, or terms is designated
as the first, another as the second, another as the third, etc.;
that is, a series is a class whose elements have been put in
one to one correspondence with some or all of the natural
numbers 1, 2, 3, - -+ (}).

5. A series may or may not have a last term, that is, it may
be finite or infinit.

For an infinit series we may state the principle of mathe-
matical induction as follows:

If the first of an infinit series of statements is tru and the truth
of any one implies the truth of the next one, then the statements
are all tru.

6. The terms of a series are frequently represented by the
name of the class to which they belong, abbreviated perhaps,
with the symbols 1, 2, 3, - - - attacht as tags. Thus a row of
houses may be called house no. 1, house no. 2, house no. 3, - - -

Frequently the terms are represented by a single letter with
the tags. Thus a row of houses may be represented by £,
hs, hs, -+-. When the figures 1, 2, 3, - - - are used in this way
they are called subscripts. The symbol % may be read
“hsub1,” etc.

7. Some of the most important series are series of numbers;
as, for example,

the series of the natural numbers 1, 2, 3,4, 5,6, ---;
the series of the squares of the natural numbers 12, 22, 3%, 42,
5%, 62, -+

the series of odd numbers 1, 3,5,7,9, +--;

(1) For a broader use of the word ‘‘series” see Edward V. Huntington,
The Continuum as a Type of Order, Annals of Mathematics, Second Series,
Vol. 6, No. 4, July 1905.