116
II - Convergence: Discrete variables
bijection -, is not so identified: this will be the least upper bound 32 of the
set of its unordered partial sums (2). If, as suggested above, one chooses a
bijection f of N onto I arbitrarily, which transforms the sum (1) into the
classical series u(f(n)) and, consequently, the unordered partial sums (2) of
the series (1) into those of this series, it is clear that (i) the family of u(i)
has a finite unordered sum if and only if the series u(f(n)) converges, (ii)
the unordered sum (1) is equal to the sum L: u(f(n)) in the sense of nO 6.
There is thus no difference between unordered convergence and convergence
in the classical sense of the series obtained by ordering the terms of the family
(u( i) )iEI arbitrarily.
This result immediately leads to another: subjecting the terms of a series
with positive terms to an arbitrary permutation, i.e. replacing u(i) by u(f(i))
where f : I ~ I is bijective, does not change its sum, for the reason that
this operation clearly does not change the set of unordered partial sums 33 of
the series; it merely permutes them. In other words, the rule of commutativity of addition applies unrestrictedly to convergent series with positive terms,
whether indexed by N or by an arbitrary countable set I. We shall see soon
that it also applies, more generally, to "absolutely convergent" series, and to
them alone.
Let us return to the classical series indexed by N. On generalising the
argument used by Cauchy to establish the divergence of the harmonic series
(nO 7), one obtains the following result:
Cauchy's condensation criterion. Let L: u(n) be a series whose terms
tend to 0 while decreasing and put v(n) = 2n.u(2n). Then the given series is
of the same nature as the series L:v(n).
The argument of nO 7 shows that
(12.3)
v(n)j2
2n- 1 u(2n)
< u(2n-l) + u(2 n - 1 + 1) + ... + u(2n - 1) =
w(n) :::; 2 n - 1 u(2 n - 1 ) = v(n - 1)
since the block considered has 2 n - 2 n - 1 = 2 n - 1 terms whose values all lie
between u(2n-l) and u(2n). The relation v(n) :::; 2w(n) :::; 2v(n - 1) shows
that the series L: v(n) has the same nature as the series L: w(n) that one
obtains by grouping the terms of the series L: u( n) in blocks of 2 n terms; now
32 Assuming this least upper bound finite. In the opposite case, it is natural to
consider that it is the symbol +00, as we shall explain in nO 17.
33 The set E of unordered partial sums is defined as follows: x E E if and only if
there exists an F such that x = s(F). The definition is analogous for the set of
elements of a sequence, of an arbitrary family (Ui), of values of a function, etc. In
the case of a family (Ui)iEI for example, it is thus the image of I under the map
i ...... Ui. The concept of least upper bound, applied to the terms of a sequence or
to the partial sums of a series, really involves only the set of these terms.
II - Convergence: Discrete variables
bijection -, is not so identified: this will be the least upper bound 32 of the
set of its unordered partial sums (2). If, as suggested above, one chooses a
bijection f of N onto I arbitrarily, which transforms the sum (1) into the
classical series u(f(n)) and, consequently, the unordered partial sums (2) of
the series (1) into those of this series, it is clear that (i) the family of u(i)
has a finite unordered sum if and only if the series u(f(n)) converges, (ii)
the unordered sum (1) is equal to the sum L: u(f(n)) in the sense of nO 6.
There is thus no difference between unordered convergence and convergence
in the classical sense of the series obtained by ordering the terms of the family
(u( i) )iEI arbitrarily.
This result immediately leads to another: subjecting the terms of a series
with positive terms to an arbitrary permutation, i.e. replacing u(i) by u(f(i))
where f : I ~ I is bijective, does not change its sum, for the reason that
this operation clearly does not change the set of unordered partial sums 33 of
the series; it merely permutes them. In other words, the rule of commutativity of addition applies unrestrictedly to convergent series with positive terms,
whether indexed by N or by an arbitrary countable set I. We shall see soon
that it also applies, more generally, to "absolutely convergent" series, and to
them alone.
Let us return to the classical series indexed by N. On generalising the
argument used by Cauchy to establish the divergence of the harmonic series
(nO 7), one obtains the following result:
Cauchy's condensation criterion. Let L: u(n) be a series whose terms
tend to 0 while decreasing and put v(n) = 2n.u(2n). Then the given series is
of the same nature as the series L:v(n).
The argument of nO 7 shows that
(12.3)
v(n)j2
2n- 1 u(2n)
< u(2n-l) + u(2 n - 1 + 1) + ... + u(2n - 1) =
w(n) :::; 2 n - 1 u(2 n - 1 ) = v(n - 1)
since the block considered has 2 n - 2 n - 1 = 2 n - 1 terms whose values all lie
between u(2n-l) and u(2n). The relation v(n) :::; 2w(n) :::; 2v(n - 1) shows
that the series L: v(n) has the same nature as the series L: w(n) that one
obtains by grouping the terms of the series L: u( n) in blocks of 2 n terms; now
32 Assuming this least upper bound finite. In the opposite case, it is natural to
consider that it is the symbol +00, as we shall explain in nO 17.
33 The set E of unordered partial sums is defined as follows: x E E if and only if
there exists an F such that x = s(F). The definition is analogous for the set of
elements of a sequence, of an arbitrary family (Ui), of values of a function, etc. In
the case of a family (Ui)iEI for example, it is thus the image of I under the map
i ...... Ui. The concept of least upper bound, applied to the terms of a sequence or
to the partial sums of a series, really involves only the set of these terms.