§2. Absolutely convergent series
103
that which, for all r > 0, can be approximated to within r by at least one of
the Ui·
This concept is useful in formulating the associativity of least upper
bounds: suppose that the set of indices I is the union of a family (Ij ) of
sets indexed by a set J and not necessarily disjoint; for every j E J let M j be
the least upper bound of the partial family (Ui), i E I j ; then the least upper
bound M of the full family (Ui), i E I, is equal to that of the family (Mj ),
j E J, of partial least upper bounds: this can be expressed as
(9.9)
SUPUi = sup (SUPUi).
iEI
jEJ iElj
The proof is easy. First M, majorising all the terms of the full family, majorises all those of each partial family and thus also all the M j . But there
exists, for any r > 0, an Ui > M - r; one has i E I j for some j E J, whence
M j ~ Ui for this index j and a fortiori M j ~ M - r, qed.
10 - The function log x. Roots of a positive number
Theorem 2 allows us to prove immediately some results which we will need
in any case; this and the following nO will be devoted to them.
Example 1. Consider, for x ~ 0, the sequence
(10.1)
un = 1 + xll! + x 2 /2! + ... + xn In!.
It is clearly increasing. To show that it is bounded, one remarks that, for
any number Z E C, the sequence (zn In!) is bounded (nO 5, example 6). For
z = 2x, one thus has (2x)n In! ~ M, whence xn In! ~ M/2 n and thus
by formula (6.5) for the sum of a geometric progression, qed.
This argument shows that the series
(10.2)
converges for x ~ 0. In fact, it converges for all x E C as we shall see in nO 14.
Example 2. Consider, again for x ~ 0, the sequence
(10.3)
Yn = (1 + xln)n.
The binomial formula
Yn = 1 + n(x/n)/l! + n(n - 1)(xln)2/2! + n(n - l)(n - 2)(x/n)3/3! + ...
+ n(n - 1) ... (n - n + 2)(n - n + l)(x/n)n In!
103
that which, for all r > 0, can be approximated to within r by at least one of
the Ui·
This concept is useful in formulating the associativity of least upper
bounds: suppose that the set of indices I is the union of a family (Ij ) of
sets indexed by a set J and not necessarily disjoint; for every j E J let M j be
the least upper bound of the partial family (Ui), i E I j ; then the least upper
bound M of the full family (Ui), i E I, is equal to that of the family (Mj ),
j E J, of partial least upper bounds: this can be expressed as
(9.9)
SUPUi = sup (SUPUi).
iEI
jEJ iElj
The proof is easy. First M, majorising all the terms of the full family, majorises all those of each partial family and thus also all the M j . But there
exists, for any r > 0, an Ui > M - r; one has i E I j for some j E J, whence
M j ~ Ui for this index j and a fortiori M j ~ M - r, qed.
10 - The function log x. Roots of a positive number
Theorem 2 allows us to prove immediately some results which we will need
in any case; this and the following nO will be devoted to them.
Example 1. Consider, for x ~ 0, the sequence
(10.1)
un = 1 + xll! + x 2 /2! + ... + xn In!.
It is clearly increasing. To show that it is bounded, one remarks that, for
any number Z E C, the sequence (zn In!) is bounded (nO 5, example 6). For
z = 2x, one thus has (2x)n In! ~ M, whence xn In! ~ M/2 n and thus
by formula (6.5) for the sum of a geometric progression, qed.
This argument shows that the series
(10.2)
converges for x ~ 0. In fact, it converges for all x E C as we shall see in nO 14.
Example 2. Consider, again for x ~ 0, the sequence
(10.3)
Yn = (1 + xln)n.
The binomial formula
Yn = 1 + n(x/n)/l! + n(n - 1)(xln)2/2! + n(n - l)(n - 2)(x/n)3/3! + ...
+ n(n - 1) ... (n - n + 2)(n - n + l)(x/n)n In!