§2. Absolutely convergent series
119
of the series L 8(Fp) are particular unordered partial sums of s(F); thus if
(4) converges unconditionally and has sum s, then the series 8(Fp) converges
and has sum 8' S s, whence finally 8 = s'.
So it all reduces to deciding the convergence of the classical series (7).
This follows from the comparison (6) with the series 1/pk-1 since the s(Fp)
[resp. the partial sums of (7)J are, up to factors independent of p, minorised
and majorised by the terms (resp. the partial sums) of the series L 1/pk-1.
Theorem 5 then provides us the condition for convergence, namely k > 2.
The reader can easily treat the case of the sum 36
where, this time, the summation is extended over Z3 - {O}, or to the sum
etc. The principle always remains the same, and consists of effecting regroupings of the terms to suit the problem. These sums, where the general term
depends on two, three, ... integers, are called double, triple series, etc. They
make beginners tremble with their "flood of indices" . But there is no more in
the general form (1) than in the classical theory, and when one is obliged, as
we shall be, to make it all explicit and to work in Z2 or f\j"7, one has the opportunity to familiarise oneself with the principles of Elementary Set Theory
as expounded in Chap. 1.
13 - Alternating series
Theorem 4 allows us to elucidate a class of series which, though convergent,
do not generally possess the commutativity property that we have just established for series with positive terms.
Consider for example the alternating harmonic series
(13.1)
1 - 1/2 + 1/3 - 1/4 + ...
and, first, its partial sums of odd order
S1
1,
S3
1 - (1/2 - 1/3),
S5
1 - (1/2 - 1/3) - (1/4 - 1/5),
etc. Since 1 2': 1/2 2': 1/32': ... , it is clear that they decrease while remaining
positive, since one has also, for example,
36 A particularly simple method for treating the series in two variables is to remark,
with Weierstrass, that m 2 + n 2 ;::: 2lmnl, whence a majorisation by the product
of the series ~ 1/lml s / 2 and ~ 1/lnl s / 2 and thus the condition 8 > 2. But this
method does not adapt to series in more than two variables.
119
of the series L 8(Fp) are particular unordered partial sums of s(F); thus if
(4) converges unconditionally and has sum s, then the series 8(Fp) converges
and has sum 8' S s, whence finally 8 = s'.
So it all reduces to deciding the convergence of the classical series (7).
This follows from the comparison (6) with the series 1/pk-1 since the s(Fp)
[resp. the partial sums of (7)J are, up to factors independent of p, minorised
and majorised by the terms (resp. the partial sums) of the series L 1/pk-1.
Theorem 5 then provides us the condition for convergence, namely k > 2.
The reader can easily treat the case of the sum 36
where, this time, the summation is extended over Z3 - {O}, or to the sum
etc. The principle always remains the same, and consists of effecting regroupings of the terms to suit the problem. These sums, where the general term
depends on two, three, ... integers, are called double, triple series, etc. They
make beginners tremble with their "flood of indices" . But there is no more in
the general form (1) than in the classical theory, and when one is obliged, as
we shall be, to make it all explicit and to work in Z2 or f\j"7, one has the opportunity to familiarise oneself with the principles of Elementary Set Theory
as expounded in Chap. 1.
13 - Alternating series
Theorem 4 allows us to elucidate a class of series which, though convergent,
do not generally possess the commutativity property that we have just established for series with positive terms.
Consider for example the alternating harmonic series
(13.1)
1 - 1/2 + 1/3 - 1/4 + ...
and, first, its partial sums of odd order
S1
1,
S3
1 - (1/2 - 1/3),
S5
1 - (1/2 - 1/3) - (1/4 - 1/5),
etc. Since 1 2': 1/2 2': 1/32': ... , it is clear that they decrease while remaining
positive, since one has also, for example,
36 A particularly simple method for treating the series in two variables is to remark,
with Weierstrass, that m 2 + n 2 ;::: 2lmnl, whence a majorisation by the product
of the series ~ 1/lml s / 2 and ~ 1/lnl s / 2 and thus the condition 8 > 2. But this
method does not adapt to series in more than two variables.