§2. Absolutely convergent series
123
harmonic series; this clearly amounts to saying that the series L Iunl diverges.
In fact, one can even, by reordering the terms of such a series, transform it
into a divergent series.
Take for example the case of an alternating series. The series
(13.5)
Ul + U3 +U5 + ...
of its positive terms being divergent, there is an index 2p - 1 for which the
sum of the terms of index::; 2p - 1 of (5) exceeds 10 + U2. To construct
the new series, one writes first all the terms of index::; 2p - 1 of (5), then
the term -U2, which has a new partial sum again> 10, then the terms of
index 2p + 1, ... of the series (5) until one obtains, at a rank 2q -1, a partial
sum greater than 100 + U4; one then writes the term -U4, whence a partial
sum again> 100, then the terms of index 2q + 1, ... of (5) until one obtains
a partial sum> 1000 + U6, then the term -U6, whence a new partial sum
> 1000, and so on indefinitely. It is clear that, in this way, one reorders the
initial series so that the new partial sums take arbitrarily large values, which
rules out convergence.
Historically the first known example (Dirichlet, 1837), is that of the alternating series where Un = 1/n 1 / 2 ; it becomes divergent when written in the
form
U1 + U3 - U2 + U5 + U7 - U4 + Ug + Un - U6 + ... ;
if it were indeed convergent, it would remain so if one grouped the terms in
threes; now the sum
1/(4n - 3)1/2 + 1/(4n - 1)1/2 - 1/(2n)I/2
is positive for n large (exercise!) and of the same order of magnitude as
1/n 1 / 2 ; whence divergence.
The reason for this phenomenon is simple: it is not because the terms of
the given series tend to 0 sufficiently fast that it converges - if such were
the case, the series L Iunl would converge -, it is because the terms, when
one adds them in the prescribed order, change sign often enough for the
decrease in the negative sums to compensate miraculously for the increase
in the positive sums. These compensations may evanesce when the order
of terms is upset. A construction analogous to the preceding would show
(Riemann) that, for any s E JR, one can reorder the terms so as to obtain
a convergent series with sum s. In other words, if the series L Un converges
while the series L Iunl does not, its sum can be defined only through its
ordered partial sums Ul + ... + Un. For this reason, some authors prefer to
speak of semi-convergent series.
14 - Classical absolutely convergent series
When a series E u( n) is absolutely convergent, i.e. when the series E Iu( n) I
converges, the commutativity problem does not arise: the series is a combination of convergent series with positive terms.
123
harmonic series; this clearly amounts to saying that the series L Iunl diverges.
In fact, one can even, by reordering the terms of such a series, transform it
into a divergent series.
Take for example the case of an alternating series. The series
(13.5)
Ul + U3 +U5 + ...
of its positive terms being divergent, there is an index 2p - 1 for which the
sum of the terms of index::; 2p - 1 of (5) exceeds 10 + U2. To construct
the new series, one writes first all the terms of index::; 2p - 1 of (5), then
the term -U2, which has a new partial sum again> 10, then the terms of
index 2p + 1, ... of the series (5) until one obtains, at a rank 2q -1, a partial
sum greater than 100 + U4; one then writes the term -U4, whence a partial
sum again> 100, then the terms of index 2q + 1, ... of (5) until one obtains
a partial sum> 1000 + U6, then the term -U6, whence a new partial sum
> 1000, and so on indefinitely. It is clear that, in this way, one reorders the
initial series so that the new partial sums take arbitrarily large values, which
rules out convergence.
Historically the first known example (Dirichlet, 1837), is that of the alternating series where Un = 1/n 1 / 2 ; it becomes divergent when written in the
form
U1 + U3 - U2 + U5 + U7 - U4 + Ug + Un - U6 + ... ;
if it were indeed convergent, it would remain so if one grouped the terms in
threes; now the sum
1/(4n - 3)1/2 + 1/(4n - 1)1/2 - 1/(2n)I/2
is positive for n large (exercise!) and of the same order of magnitude as
1/n 1 / 2 ; whence divergence.
The reason for this phenomenon is simple: it is not because the terms of
the given series tend to 0 sufficiently fast that it converges - if such were
the case, the series L Iunl would converge -, it is because the terms, when
one adds them in the prescribed order, change sign often enough for the
decrease in the negative sums to compensate miraculously for the increase
in the positive sums. These compensations may evanesce when the order
of terms is upset. A construction analogous to the preceding would show
(Riemann) that, for any s E JR, one can reorder the terms so as to obtain
a convergent series with sum s. In other words, if the series L Un converges
while the series L Iunl does not, its sum can be defined only through its
ordered partial sums Ul + ... + Un. For this reason, some authors prefer to
speak of semi-convergent series.
14 - Classical absolutely convergent series
When a series E u( n) is absolutely convergent, i.e. when the series E Iu( n) I
converges, the commutativity problem does not arise: the series is a combination of convergent series with positive terms.