Page 78 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 61
1
B. 1 − 2+4 − 8+16 − 32 + ··· = ,
3
1
C. 1 − 3+9 − 27+81 − 243 + ··· = .
4
1
It is clear that the sum of series B cannot be equal to 3 , since the more
terms we actually sum, the farther away the result gets from 1 3 . But the
sum of any series ought to be a limit the closer to which the partial sums
should approach, the more terms are added.

109. From this we conclude that series of this kind, which are called
divergent, have no fixed sums, since the partial sums do not approach any
limit that would be the sum for the infinite series. This is certainly a true
conclusion, since we have shown the error in neglecting the final remainder.
However, it is possible, with considerable justice, to object that these sums,
even though they seem not to be true, never lead to error. Indeed, if we
allow them, then we can discover many excellent results that we would
not have if we rejected them out of hand. Furthermore, if these sums were
really false, they would not consistently lead to true results; rather, since
they differ from the true sum not just by a small difference, but by infinity,
they should mislead us by an infinite amount. Since this does not happen,
we are left with a most difficult knot to unravel.

110. I say that the whole difficulty lies in the name sum. If, as is commonly
the case, we take the sum of a series to be the aggregate of all of its terms,
actually taken together, then there is no doubt that only infinite series
that converge continually closer to some fixed value, the more terms we
actually add, can have sums. However, divergent series, whose terms do
not decrease, whether their signs + and − alternate or not, do not really
have fixed sums, supposing we use the word sum for the aggregate of all
of the terms. Consider these cases that we have recalled, with erroneous
sums, for example the finite expression 1/ (1 − x) for the infinite series
3
2
1+x+x +x +··· . The truth of the matter is this, not that the expression
is the sum of the series, but that the series is derived from the expression.
In this situation the name sum could be completely omitted.
111. These inconveniences and apparent contradictions can be avoided
if we give the word sum a meaning different from the usual. Let us say
that the sum of any infinite series is a finite expression from which the
series can be derived. In this sense, the true sum of the infinite series
2 3
1+x+x +x +··· is 1/ (1 − x), since this series is derived from the fraction,
no matter what value is substituted for x. With this understanding, if
the series is convergent, the new definition of sum agrees with the usual
definition. Since divergent series do not have a sum, properly speaking,
there is no real difficulty which arises from this new meaning. Finally, with
the aid of this definition we can keep the usefulness of divergent series and
preserve their reputations.