Rearrangements of numerical series
Marion Scheepers
October 13, 2011
Marion Scheepers
Rearrangements of numerical series
Notation, conventions
Signwise monotonic
a1 , a2 , · · · , an , · · · :
−b1 , −b2 , · · · , −bn , · · · :
Marion Scheepers
Rearrangements of numerical series
f : N −→ R
Positive terms of f in order.
Negative terms of f in order
Nicolas Oresme’s Theorem (1320 - 1382)
Theorem (Oresme)
The series
∞
!
1
n=1
is divergent.
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Rearrangements of numerical series
n
The Leibniz Convergence Test (1675)
Theorem (Leibniz)
If (an : n = 1, 2, 3, ...) is a monotonic sequence of real numbers
such that limn→∞ an = 0, then the series
∞
!
n=1
is convergent.
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Rearrangements of numerical series
(−1)n−1 an
Thus, each of the series
∞
!
(−1)n−1
n=1
∞
!
(−1)n−1
n=1
and
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Rearrangements of numerical series
√
n
∞
!
(−1)n
n=2
is conditionally convergent.
n
n ln(n)
,
Dirichlet’s Observations (1837)
The rearrangement
1 1 1 1 1 1
+ − + + − + ···
1 3 2 5 7 4
converges, while the rearrangement
1
1
1
1
1
1
√ + √ − √ + √ + √ − √ + ···
5
7
4
1
3
2
diverges.
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Rearrangements of numerical series
Martin Ohm’s Theorem (1839)
Theorem (M. Ohm)
n−1
For p and q positive integers rearrange ( (−1)n
: n = 1, 2, · · · ) by
taking the first p positive terms, then the first q negative terms,
then the next p positive terms, then the next q negative terms, and
so on. The rearranged series converges to
ln(2) +
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Rearrangements of numerical series
1
p
ln( ).
2
q
Riemann’s Theorem (1854)
Theorem (Riemann)
"
A numerical series f is conditionally convergent if, and only if,
there is for each real number α a rearrangement of this series
which converges to α.
Marion Scheepers
Rearrangements of numerical series
Observations
The rearrangement
1 1 1 1 1 1
+ − + + − + ···
1 3 2 5 7 4
converges to a different sum than
rearrangement
"∞
n=1
(−1)n−1
,
n
while the
1
1
1
1
1
1
+
−
+
+
−
+ ···
2 ln(2) 4 ln(4) 3 ln(3) 6 ln(6) 8 ln(8) 5 ln(5)
converges to the same sum as
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Rearrangements of numerical series
"∞ (−1)n
2 n ln(n) .
Schlömilch’s Theorem (1873)
Theorem (Schlömilch)
"
Let f be signwise monotonic and
f conditionally convergent. For
p and q positive integers rearrange f by taking the first p positive
terms, then the first q negative terms, and so on. The rearranged
series converges to
∞
!
p
f (n) + g ln( )
q
n=1
where g is the limit limn→∞ n · an .
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Rearrangements of numerical series
Asymptotic density
A ⊆ N, n ∈ N
πA (n)
d(A)
=
=
|{x ∈ A : x ≤ n}|
limn→∞ πAn(n)
d(A) is the asymptotic density of A when this limit exists.
fA (n) =
#
aj
−bj
if n is the j-th element of A.
if n is the j-th element of N \ A.
ωf = {x ∈ (0, 1) : (∃A ⊆ N)(d(A) = x and
σf = {x ∈ (0, 1) : (∀A ⊆ N)(d(A) = x and
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Rearrangements of numerical series
!
!
fA converges)}
fA converges)}
Pringsheim’s Theorems (1883)
Pringsheim found:
"
A) Convergence criteria of fB when lim n · an = ∞.
"
B) Convergence criteria of fB when lim n · an = 0.
"
C) The change in value of fB for all B with 0 < d(B) < 1 when
lim n · an = g *= 0.
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Rearrangements of numerical series
Regarding Pringsheim’s Theorem A)
Theorem
Let f be signwise monotonic, converging to 0. Let 0 < x < 1 be
given. The following are equivalent:
1
2
x ∈ ωf , and lim n · an = ∞.
For each set B such that
ωf = {x }.
Note: In this case σf = ∅.
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Rearrangements of numerical series
"
fB converges, d(B) = x (i.e.,
A Lemma
Lemma
Let f be signwise monotonic. If |ωf | > 1, then for all A, B ⊆ N
"
"
such that d(A) = d(B) and fA converges, also fB converges,
"
"
and fA = fB .
In this case
Φf (x ) =
!
fA , A some subset of N with d(A) = x
is independent of the choice of A.
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Rearrangements of numerical series
Regarding Pringsheim’s Theorem B)
Theorem
Let f be signwise monotonic, converging to 0. Let x ∈ R be given.
The following are equivalent:
1
2
3
4
ω(f ) ∩ (0, 1) *= ∅, and lim n · an = 0.
For each set B such that 0 < d(B) < 1,
"
fB converges to x .
ωf ⊇ (0, 1) and Φf is constant of value x on (0, 1).
ωf = [0, 1].
In this case, σf = (0, 1).
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Rearrangements of numerical series
Regarding Pringsheim’s Theorem C)
Theorem
Let f be signwise monotonic. Let x ∈ R be given. The following
are equivalent:
1
2
3
ωf is dense in some interval.
σf = (0, 1).
lim n · an exists and for all x , y in (0, 1),
Φf (x ) = Φf (y ) + lim n · an ln(
.
In this case, ωf = (0, 1).
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Rearrangements of numerical series
x (1 − y )
)
y (1 − x )
A detour to groups
For x, y in (0,1), define
x .y =
xy
.
1 − x − y + 2xy
Fact 1: ((0, 1), .) is an Abelian group with identity element 21 .
For g a positive real define
Ψg : (0, 1) −→ R : x /→ g ln(
x
).
1−x
Fact 2: Ψg is a group isomorphism from ((0, 1), .) to (R, +).
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Rearrangements of numerical series
Return to Pringsheim’s Theorem C)
Let f be signwise monotonic with σf = (0, 1) and Φf
non-constant. Put g = lim n · an . Then g > 0.
1
Φf (·) − Φf ( ) : (σf , .) −→ (R, +)
2
is a group isomorphism.
The function
x (1 − y )
d(x , y ) = g| ln(
)|
y (1 − x )
is a metric on σf , and measures |
and d(B).
Marion Scheepers
Rearrangements of numerical series
"
fA −
"
fB | in terms of d(A)