§2. Absolutely convergent series
131
The inequality, where one replaces the u and the v by their moduli, is
certainly valid if one restricts the sums to a finite subset F of the set I of
indices (Chap. III, Appendix). It is a fortiori valid if one sums the left hand
side over F and the right hand side over I. The squares of the partial sums
of the series L IU{Vil are thus bounded above by the right hand side of the
relation to be established, qed.
To finish this study of unconditional convergence for the time being -
the law of associativity remains to be proved, nO 18 -, let us return to the
absolutely convergent classical series of the preceding nO. By definition, the
series L lu(n)1 converges; it therefore satisfies (6), and therefore the given
series L u(n) converges unconditionally, and vice-versa.
Thus, for the classical ordered series, there is no difference between unconditional convergence and absolute convergence as in n° 14.
We have, however, to show that the classical definition of the sum
s = lim (u(O) + ... + u(n))
provides the same result as definition (4) of the unordered sum s(N), supposing that the series starts with the term u(O). To do this let us choose a
number r > 0 and a finite subset F of N satisfying (4). Since F is finite, the set
Fn = {O, 1, ... , n} contains F for n sufficiently large. So Is(Fn) - s(N) I ::; r
for n large. But s(Fn), the sum of n first terms of the series, is equal to s to
within r for n large; whence Is - s(N)1 ::; 2r, qed.
Finally, consider an absolutely summable family (u(i)), i E I, and choose
a bijection f of N onto I. The unordered sums of the given family and those
of the series L u(f(n)) are clearly the same, and similarly for those obtained by replacing the u( i) by their absolute values. It follows that the
series L u(f (n)) is absolutely convergent and has the same sum as the given
family. If, conversely, a bijection of N onto I transforms the family (u( i)) into
an absolutely convergent series, it is clear that the given family is absolutely
summable. This shows that there is in fact no difference in nature between
general unconditional convergence and classical absolute convergence.
The interest of unconditional convergence will appear later, when after
establishing associativity, we will apply it to multiple series for which the
choice of a bijection f does not provide any result, except by a miracle 4o . For
the moment, we shall return to more classical, not to say more hackneyed,
aspects of the traditional theory of series.
40 Hardly surprising. If you subject the terms of a geometric series, for example, to
an arbitrary permutation, you will have severe difficulties in demonstrating the
convergence of the new series using only its ordered partial sums - unless you
reconstitute an ad hoc proof of the general theorem.
131
The inequality, where one replaces the u and the v by their moduli, is
certainly valid if one restricts the sums to a finite subset F of the set I of
indices (Chap. III, Appendix). It is a fortiori valid if one sums the left hand
side over F and the right hand side over I. The squares of the partial sums
of the series L IU{Vil are thus bounded above by the right hand side of the
relation to be established, qed.
To finish this study of unconditional convergence for the time being -
the law of associativity remains to be proved, nO 18 -, let us return to the
absolutely convergent classical series of the preceding nO. By definition, the
series L lu(n)1 converges; it therefore satisfies (6), and therefore the given
series L u(n) converges unconditionally, and vice-versa.
Thus, for the classical ordered series, there is no difference between unconditional convergence and absolute convergence as in n° 14.
We have, however, to show that the classical definition of the sum
s = lim (u(O) + ... + u(n))
provides the same result as definition (4) of the unordered sum s(N), supposing that the series starts with the term u(O). To do this let us choose a
number r > 0 and a finite subset F of N satisfying (4). Since F is finite, the set
Fn = {O, 1, ... , n} contains F for n sufficiently large. So Is(Fn) - s(N) I ::; r
for n large. But s(Fn), the sum of n first terms of the series, is equal to s to
within r for n large; whence Is - s(N)1 ::; 2r, qed.
Finally, consider an absolutely summable family (u(i)), i E I, and choose
a bijection f of N onto I. The unordered sums of the given family and those
of the series L u(f(n)) are clearly the same, and similarly for those obtained by replacing the u( i) by their absolute values. It follows that the
series L u(f (n)) is absolutely convergent and has the same sum as the given
family. If, conversely, a bijection of N onto I transforms the family (u( i)) into
an absolutely convergent series, it is clear that the given family is absolutely
summable. This shows that there is in fact no difference in nature between
general unconditional convergence and classical absolute convergence.
The interest of unconditional convergence will appear later, when after
establishing associativity, we will apply it to multiple series for which the
choice of a bijection f does not provide any result, except by a miracle 4o . For
the moment, we shall return to more classical, not to say more hackneyed,
aspects of the traditional theory of series.
40 Hardly surprising. If you subject the terms of a geometric series, for example, to
an arbitrary permutation, you will have severe difficulties in demonstrating the
convergence of the new series using only its ordered partial sums - unless you
reconstitute an ad hoc proof of the general theorem.