Rearrangements of numerical series Marion Scheepers October 13, - - PowerPoint PPT Presentation

Rearrangements of numerical series

Marion Scheepers October 13, 2011

Marion Scheepers Rearrangements of numerical series

Notation, conventions

f : N − → R Signwise monotonic a1, a2, · · · , an, · · · : Positive terms of f in order. −b1, −b2, · · · , −bn, · · · : Negative terms of f in order

Marion Scheepers Rearrangements of numerical series

Nicolas Oresme’s Theorem (1320 - 1382)

Theorem (Oresme) The series

∞

• n=1

1 n is divergent.

Marion Scheepers Rearrangements of numerical series

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

(−1)n−1an is convergent.

Marion Scheepers Rearrangements of numerical series

Thus, each of the series

∞

• n=1

(−1)n−1 n ,

∞

• n=1

(−1)n−1 √n and

∞

• n=2

(−1)n n ln(n) is conditionally convergent.

Marion Scheepers Rearrangements of numerical series

Dirichlet’s Observations (1837)

The rearrangement 1 1 + 1 3 − 1 2 + 1 5 + 1 7 − 1 4 + · · · converges, while the rearrangement 1 √ 1 + 1 √ 3 − 1 √ 2 + 1 √ 5 + 1 √ 7 − 1 √ 4 + · · · diverges.

Marion Scheepers Rearrangements of numerical series

Martin Ohm’s Theorem (1839)

Theorem (M. Ohm) For p and q positive integers rearrange ((−1)n−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) + 1 2 ln(p q ).

Marion Scheepers Rearrangements of numerical series

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 3 − 1 2 + 1 5 + 1 7 − 1 4 + · · · converges to a different sum than ∞

n=1 (−1)n−1 n

, while the rearrangement 1 2 ln(2) + 1 4 ln(4) − 1 3 ln(3) + 1 6 ln(6) + 1 8 ln(8) − 1 5 ln(5) + · · · converges to the same sum as ∞

2 (−1)n n ln(n).

Marion Scheepers Rearrangements of numerical series

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

∞

• n=1

f (n) + g ln(p q ) where g is the limit limn→∞ n · an.

Marion Scheepers Rearrangements of numerical series

Asymptotic density

A ⊆ N, n ∈ N πA(n) = |{x ∈ A : x ≤ n}| d(A) = limn→∞

πA(n) n

d(A) is the asymptotic density of A when this limit exists. fA(n) =

• aj

if n is the j-th element of A. −bj if n is the j-th element of N \ A. ωf = {x ∈ (0, 1) : (∃A ⊆ N)(d(A) = x and

• fA converges)}

σf = {x ∈ (0, 1) : (∀A ⊆ N)(d(A) = x and

• fA converges)}

Marion Scheepers Rearrangements of numerical series

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.

Marion Scheepers 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 x ∈ ωf , and lim n · an = ∞. 2 For each set B such that fB converges, d(B) = x (i.e.,

ωf = {x}. Note: In this case σf = ∅.

Marion Scheepers Rearrangements of numerical series

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.

Marion Scheepers 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 ω(f ) ∩ (0, 1) = ∅, and lim n · an = 0. 2 For each set B such that 0 < d(B) < 1, fB converges to x. 3 ωf ⊇ (0, 1) and Φf is constant of value x on (0, 1). 4 ωf = [0, 1].

In this case, σf = (0, 1).

Marion Scheepers 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 ωf is dense in some interval. 2 σf = (0, 1). 3 lim n · an exists and for all x, y in (0, 1),

Φf (x) = Φf (y) + lim n · an ln(x(1 − y) y(1 − x)) . In this case, ωf = (0, 1).

Marion Scheepers Rearrangements of numerical series

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 1

2.

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, +).

Marion Scheepers 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. Φf (·) − Φf (1 2) : (σf , ⊙) − → (R, +) is a group isomorphism. The function d(x, y) = g| ln(x(1 − y) y(1 − x))| is a metric on σf , and measures | fA − fB| in terms of d(A) and d(B).

Marion Scheepers Rearrangements of numerical series