118
II - Convergence: Discrete variables
chosen pn for any integer p > 1, considering that there are (p -l)pn integers
between pn and pn+l, so that the inequalities (3) remain valid in this case
after trivial modifications. In fact, Cauchy must surely have observed that
the criterion with pn applies precisely to the same series as the criterion with
2 n so he must have thought it better to leave to others, if by chance there
were such people, the glory of a futile generalisation.
Theorem 5 admits generalisations to multiple series, which will allow us
to understand the utility of "unconditional convergence" introduced above.
Consider for example the sum
(12.4)
L 1/(m 2 + n 2 / 12 = L u(m, n)
(m,n)EZ2-{O}
mentioned in nO 6. Its resemblance to the series
f l/n k = ~ L 1/(n 2 / 12
n=l
nEZ-{O}
(12.5)
of Theorem 5 is sufficiently striking for one to hope for a similar result. More
to explain the general method than to press on further, let us consider for
every integer p ~ 1 the set Fp C I = 7!..2 - {O} of pairs (m, n) such that
Iml + Inl = p. There are 4p, apart from error or omission (diagram!). For
such a pair one has
p2/4 :$ m 2 + n 2 :$ 2p2
since each of the two integers Iml, Inl is :$ p and one of the two is ~ p/2; the
partial sum corresponding to Fp thus satisfies the relation
(12.6)
with constants m, M > 0 independent of p and whose exact values matter
little. Since the set of indices I is the union of the pairwise disjoint sets Fp
one can assume - associativity ... - that the convergence of (4) is governed
by that of the classical series Es(Fp). Let us first justify this point. Let F
be any finite subset of I. It is the union of the sets FnFp, which are pairwise
disjoint, and only a finite number of them are not empty, since F is finite.
One thus has s(F) = E s(F n Fp) :$ E s(Fp) since all the u(i) are positive.
The convergence of the series s(Fp) then shows the existence of a number
which majorises all the partial sums s(F), namely
(12.7)
so the sum (4) converges unconditionally, with a total sum s :::; s'. Conversely,
the ordered partial sums
II - Convergence: Discrete variables
chosen pn for any integer p > 1, considering that there are (p -l)pn integers
between pn and pn+l, so that the inequalities (3) remain valid in this case
after trivial modifications. In fact, Cauchy must surely have observed that
the criterion with pn applies precisely to the same series as the criterion with
2 n so he must have thought it better to leave to others, if by chance there
were such people, the glory of a futile generalisation.
Theorem 5 admits generalisations to multiple series, which will allow us
to understand the utility of "unconditional convergence" introduced above.
Consider for example the sum
(12.4)
L 1/(m 2 + n 2 / 12 = L u(m, n)
(m,n)EZ2-{O}
mentioned in nO 6. Its resemblance to the series
f l/n k = ~ L 1/(n 2 / 12
n=l
nEZ-{O}
(12.5)
of Theorem 5 is sufficiently striking for one to hope for a similar result. More
to explain the general method than to press on further, let us consider for
every integer p ~ 1 the set Fp C I = 7!..2 - {O} of pairs (m, n) such that
Iml + Inl = p. There are 4p, apart from error or omission (diagram!). For
such a pair one has
p2/4 :$ m 2 + n 2 :$ 2p2
since each of the two integers Iml, Inl is :$ p and one of the two is ~ p/2; the
partial sum corresponding to Fp thus satisfies the relation
(12.6)
with constants m, M > 0 independent of p and whose exact values matter
little. Since the set of indices I is the union of the pairwise disjoint sets Fp
one can assume - associativity ... - that the convergence of (4) is governed
by that of the classical series Es(Fp). Let us first justify this point. Let F
be any finite subset of I. It is the union of the sets FnFp, which are pairwise
disjoint, and only a finite number of them are not empty, since F is finite.
One thus has s(F) = E s(F n Fp) :$ E s(Fp) since all the u(i) are positive.
The convergence of the series s(Fp) then shows the existence of a number
which majorises all the partial sums s(F), namely
(12.7)
so the sum (4) converges unconditionally, with a total sum s :::; s'. Conversely,
the ordered partial sums