ABSTRACT
This result allows us to define various more general infinite sums. In general we shall write the normal form of a number x in the form £ co*. r^ ,
i it being understood that the sum is over all y e No, and that the numbers ry satisfy the required conditions. If now we have a set or sequence of numbers xH = £ co*. rH y, then we say that the sum £ x, is convergent to x (in some
y ' " sense) if and only if all the real number sums are convergent (in the
n * .same sense) to sums r , say, and x is the number Y,co> .ry, and furthermore all the rH J vanish for all y not in some descending sequence ( y ^ fi < a). This last restriction is quite essential to prevent certain absurdities-without it we should have
(1 - to) + (co — a)2) + (co2 - co3) + ... = 1,
in which an infinite sequence of negative numbers has positive sum. We call a number infinitesimal if it lies between — r and r for every positive real number r.
