§2. Absolutely convergent series
139
(IV) If lim Un = +00 and if Vn tends to a finite limit or to +00, then lim(un +
vn ) = +00. It is even enough that the sequence (vn ) be bounded below:
if indeed Vn > B for all n, the relation Un + Vn > A is satisfied once
Un> A - B, SO for n large.
On the other hand one can say nothing when Un and Vn tend respectively
to +00 and -00, as trivial examples show: n 2 tends to +00, -n tends to -00,
but n 2 - n tends to +00; n 2 tends to +00, -2n2 tends to -00, but n 2 - 2n 2
tends to -00; n tends to +00, -n + sin n tends to -00, but sin n has no limit.
Whence a great mystery which has mystified a number of mystics meditating
on the infinite: the difference (+00) - (+00), which some write 00 - 00, is
meaningless even though one can always claim that the relations
( +00) _ 10100000000
(-00)+(-00)
+00, (+00) + (+00) = 00,
-00, (+00) - (-00) = +00
have a meaning and are even correct if one states precisely what they mean:
back to Hardy again ...
(V) If lim Un = +00 and if Vn tends to +00 or to a strictly positive limit,
then lim UnVn = +00. Indeed, there is a number m > 0 such that one
has Vn > m for n large, so UnVn > m.un since all are> 0 for n large,
and it remains to apply the rules (I) and (II) - or we may write that
Un > A/m implies UnVn > A.
Part (V) remains valid, with a change of sign in the limit, if Vn tends
to -00 or to a finite strictly negative limit.
If on the other hand Vn tends to 0, anything may happen: n 2 .l/n tends
to +00; n 2 .1/3n 2 tends to 1/3; n 2 .1/n 3 tends to 0; and n 2 .sinn/n 2 has no
limit. Whence a new mystery: the product 0.00 is meaningless, as clearly
is the so-called quotient 00/00, as shown by the preceding examples. One
cannot deduce anything without imposing much more precise hypotheses on
the behaviour of the sequences Un and Vn as n increases indefinitely. This is
one of the aims of the theory of asymptotic expansions of Chap. VI.
18 - Unconditional convergence: associativity
As we saw in nO 12 d propos the sum L: 1/(m 2 +n 2 )k/2, it can be helpful, when
determining whether such an expression converges, to regroup the terms,
though not by a simplistic procedure, as described in nO 12, where one groups
in their natural order the terms of a series indexed by N. We shall now show
that this type of operation is allowable unrestrictedly when applied to sums
which converge unconditionally.
Let us start from a family (u(i)), i E I, where I is countable. To group
the terms of the sum L: u( i) is to construct a partition
139
(IV) If lim Un = +00 and if Vn tends to a finite limit or to +00, then lim(un +
vn ) = +00. It is even enough that the sequence (vn ) be bounded below:
if indeed Vn > B for all n, the relation Un + Vn > A is satisfied once
Un> A - B, SO for n large.
On the other hand one can say nothing when Un and Vn tend respectively
to +00 and -00, as trivial examples show: n 2 tends to +00, -n tends to -00,
but n 2 - n tends to +00; n 2 tends to +00, -2n2 tends to -00, but n 2 - 2n 2
tends to -00; n tends to +00, -n + sin n tends to -00, but sin n has no limit.
Whence a great mystery which has mystified a number of mystics meditating
on the infinite: the difference (+00) - (+00), which some write 00 - 00, is
meaningless even though one can always claim that the relations
( +00) _ 10100000000
(-00)+(-00)
+00, (+00) + (+00) = 00,
-00, (+00) - (-00) = +00
have a meaning and are even correct if one states precisely what they mean:
back to Hardy again ...
(V) If lim Un = +00 and if Vn tends to +00 or to a strictly positive limit,
then lim UnVn = +00. Indeed, there is a number m > 0 such that one
has Vn > m for n large, so UnVn > m.un since all are> 0 for n large,
and it remains to apply the rules (I) and (II) - or we may write that
Un > A/m implies UnVn > A.
Part (V) remains valid, with a change of sign in the limit, if Vn tends
to -00 or to a finite strictly negative limit.
If on the other hand Vn tends to 0, anything may happen: n 2 .l/n tends
to +00; n 2 .1/3n 2 tends to 1/3; n 2 .1/n 3 tends to 0; and n 2 .sinn/n 2 has no
limit. Whence a new mystery: the product 0.00 is meaningless, as clearly
is the so-called quotient 00/00, as shown by the preceding examples. One
cannot deduce anything without imposing much more precise hypotheses on
the behaviour of the sequences Un and Vn as n increases indefinitely. This is
one of the aims of the theory of asymptotic expansions of Chap. VI.
18 - Unconditional convergence: associativity
As we saw in nO 12 d propos the sum L: 1/(m 2 +n 2 )k/2, it can be helpful, when
determining whether such an expression converges, to regroup the terms,
though not by a simplistic procedure, as described in nO 12, where one groups
in their natural order the terms of a series indexed by N. We shall now show
that this type of operation is allowable unrestrictedly when applied to sums
which converge unconditionally.
Let us start from a family (u(i)), i E I, where I is countable. To group
the terms of the sum L: u( i) is to construct a partition