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78 THE NATURE OF MATHEMATICS 

use of infinite series as a means of approximate calculation. I 
shall distinguish what I call "sequences" and "series." A 
sequence is a collection finite or infinite of numbers; a 
series is a finite or infinite collection of numbers connected by 
addition. Sequences and series can be made to correspond in 
the following way. To the sequence 1, 2, 3, 4, . . . belongs 
a series of which the terms are got by subtracting, in order, 
the terms of the sequence from the ones immediately following 
them, thus : 

(2_l) + (3-2) + (4-3) + . . . = 1 + 1 + 1+ . . .; 
and from a series a corresponding sequence can be got by making 
the sum of the first, the first two, the first three, . . . terms 
the first, second, third, . . . term of the sequence respectively. 
Thus, to the series 1 + 1 + 1 + . . . corresponds the sequence 
1, 2, 3, ... 

Now, if a series has only a finite number of terms, it is 
possible to find the sum of all the terms ; but if the series is 
unending, we evidently cannot. But in certain cases the 
corresponding sequence has a limit, and this limit is called by 
mathematicians, neither unnaturally nor accurately, " the sum 
to infinity of the series." Thus, the sequence 1, 1 + J, 1 + 1 
+ J,... has the limit 2, and so the sum to infinity of the 
series 1 + J + i + f + ... is 2. Of course, all series do not 
have a sum : thus 1 + 1 + 1 +. . . to infinity, has not the 
terms of the corresponding sequence increase continually 
beyond all limits. Notice particularly that the terms of a 
sequence may increase continually, and yet have a limit 
those of the above sequence with limit 2 so increase, but not 
beyond 2, though they do beyond any number less than 2 ; 
also notice that the terms of a sequence may increase beyond 
all limits even if the terms of the corresponding series con- 
tinually diminish, remaining positive, towards 0. The series 
l + i + + + ^- + i 8 sucn a series ; the terms of the 
sequence slowly increase beyond all limits, as we see when 
we reflect that the sums 



i are all greater than . It is very important to realise the 
fact illustrated by this example; for it shows that the con- 
ditions under which an infinite series has a sum are by no 
means as simple as they might appear at first sight.