§2. Absolutely convergent series
115
of the initial series, qed. We will show further that one can even perform
regroupings of infinitely many terms and permute the terms of the series arbitrarily without changing the result, which is "obvious" only so long as one
has not understood the difference between algebra (finite sums) and analysis
(infinite sums).
We said at the beginning of n° 6 that the fundamental problem of the
theory of series would be to give a meaning to the sum of an infinite family
(U(i))iEI of real or complex numbers indexed by any countable set I, for
example I = N x N = N 2 , or N x N x N = N 3 , etc. The preceding arguments
already allow us to treat the case of a sum of positive numbers.
An "obvious" method to give a meaning to the expression
(12.1)
Lu(i)
iEI
would be to choose a bijection f : N ~ I, to put
v(n) = u(J(n))
and to declare that the sum (1) is, by definition, equal to the sum of the
series v(n). This would presume that we had established that the result is
independent of the choice of f, and since two bijections f and 9 of I onto N
differ from one another by a bijection of N onto N, Le. by a permutation of
N, this reduces to showing that the sum of a convergent series with positive
terms is independent of the order of its terms. This is indeed the case, but
one can in fact proceed directly, Le. without choosing a particular bijection
of N onto I.
Let us take inspiration from the case where I = N. The sum of the series
is then the least upper bound of the set of its partial sums, i.e. the least
number which majorises them. But these are the ordered partial sums and
not the unordered partial sums
(1.2.2)
s(F) = L u(i)
iEF
obtained by adding those terms of the series whose index belongs to an arbitrary finite subset F of the set of indices I = N. If nevertheless you add
the terms of index 1492, 1776, 1812, 1861, 1898, 1917 and 1941 the fact that
the series has positive terms shows that the result is :::; u(l) + u(2) + ... +
u(1940) + u(1941). A number which majorises all the ordered partial sums
thus also majorises all the unordered partial sums, and conversely, since the
ordered partial sums appear among the unordered partial sums.
We thus might also define the sum of a series with positive terms as the
least upper bound of the set of its unordered partial sums.
It is now easy to define directly the "unconditional" sum (hint: total) (1)
in the general case where the set I, though identifiable with N - choose a
115
of the initial series, qed. We will show further that one can even perform
regroupings of infinitely many terms and permute the terms of the series arbitrarily without changing the result, which is "obvious" only so long as one
has not understood the difference between algebra (finite sums) and analysis
(infinite sums).
We said at the beginning of n° 6 that the fundamental problem of the
theory of series would be to give a meaning to the sum of an infinite family
(U(i))iEI of real or complex numbers indexed by any countable set I, for
example I = N x N = N 2 , or N x N x N = N 3 , etc. The preceding arguments
already allow us to treat the case of a sum of positive numbers.
An "obvious" method to give a meaning to the expression
(12.1)
Lu(i)
iEI
would be to choose a bijection f : N ~ I, to put
v(n) = u(J(n))
and to declare that the sum (1) is, by definition, equal to the sum of the
series v(n). This would presume that we had established that the result is
independent of the choice of f, and since two bijections f and 9 of I onto N
differ from one another by a bijection of N onto N, Le. by a permutation of
N, this reduces to showing that the sum of a convergent series with positive
terms is independent of the order of its terms. This is indeed the case, but
one can in fact proceed directly, Le. without choosing a particular bijection
of N onto I.
Let us take inspiration from the case where I = N. The sum of the series
is then the least upper bound of the set of its partial sums, i.e. the least
number which majorises them. But these are the ordered partial sums and
not the unordered partial sums
(1.2.2)
s(F) = L u(i)
iEF
obtained by adding those terms of the series whose index belongs to an arbitrary finite subset F of the set of indices I = N. If nevertheless you add
the terms of index 1492, 1776, 1812, 1861, 1898, 1917 and 1941 the fact that
the series has positive terms shows that the result is :::; u(l) + u(2) + ... +
u(1940) + u(1941). A number which majorises all the ordered partial sums
thus also majorises all the unordered partial sums, and conversely, since the
ordered partial sums appear among the unordered partial sums.
We thus might also define the sum of a series with positive terms as the
least upper bound of the set of its unordered partial sums.
It is now easy to define directly the "unconditional" sum (hint: total) (1)
in the general case where the set I, though identifiable with N - choose a