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 a 1 , a 2 , · · · , a n , · · · : Positive terms of f in order. − b 1 , − b 2 , · · · , − b n , · · · : Negative terms of f in order Marion Scheepers Rearrangements of numerical series

Nicolas Oresme’s Theorem (1320 - 1382) Theorem (Oresme) The series ∞ 1 � n n = 1 is divergent. Marion Scheepers Rearrangements of numerical series

The Leibniz Convergence Test (1675) Theorem (Leibniz) If ( a n : n = 1 , 2 , 3 , ... ) is a monotonic sequence of real numbers such that lim n →∞ a n = 0 , then the series ∞ ( − 1 ) n − 1 a n � n = 1 is convergent. Marion Scheepers Rearrangements of numerical series

Thus, each of the series ( − 1 ) n − 1 ∞ � , n n = 1 ∞ ( − 1 ) n − 1 � √ n n = 1 and ( − 1 ) n ∞ � n ln ( n ) n = 2 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 + 1 5 + 1 7 − 1 √ √ √ √ √ √ 4 + · · · 1 3 2 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 = 1 , 2 , · · · ) by n 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 + · · · ( − 1 ) n − 1 converges to a different sum than � ∞ , while the n = 1 n rearrangement 1 1 1 1 1 1 2 ln ( 2 ) + 4 ln ( 4 ) − 3 ln ( 3 ) + 6 ln ( 6 ) + 8 ln ( 8 ) − 5 ln ( 5 ) + · · · ( − 1 ) n converges to the same sum as � ∞ n ln ( n ) . 2 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 ∞ f ( n ) + g ln ( p � q ) n = 1 where g is the limit lim n →∞ n · a n . Marion Scheepers Rearrangements of numerical series

Asymptotic density A ⊆ N , n ∈ N π A ( n ) = |{ x ∈ A : x ≤ n }| π A ( n ) d ( A ) = lim n →∞ n d ( A ) is the asymptotic density of A when this limit exists. � a j if n is the j-th element of A . f A ( n ) = − b j if n is the j-th element of N \ A . � ω f = { x ∈ ( 0 , 1 ) : ( ∃ A ⊆ N )( d ( A ) = x and f A converges ) } � σ f = { x ∈ ( 0 , 1 ) : ( ∀ A ⊆ N )( d ( A ) = x and f A converges ) } Marion Scheepers Rearrangements of numerical series

Pringsheim’s Theorems (1883) Pringsheim found: A) Convergence criteria of � f B when lim n · a n = ∞ . B) Convergence criteria of � f B when lim n · a n = 0. C) The change in value of � f B for all B with 0 < d ( B ) < 1 when lim n · a n = 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 ∈ ω f , and lim n · a n = ∞ . 2 For each set B such that � f B converges, d ( B ) = x (i.e., ω f = { x } . Note: In this case σ f = ∅ . Marion Scheepers Rearrangements of numerical series

A Lemma Lemma Let f be signwise monotonic. If | ω f | > 1 , then for all A , B ⊆ N such that d ( A ) = d ( B ) and � f A converges, also � f B converges, and � f A = � f B . In this case � Φ f ( x ) = f A , 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 · a n = 0 . 2 For each set B such that 0 < d ( B ) < 1 , � f B converges to x. 3 ω f ⊇ ( 0 , 1 ) and Φ f is constant of value x on ( 0 , 1 ) . 4 ω f = [ 0 , 1 ] . In this case, σ f = ( 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 ω f is dense in some interval. 2 σ f = ( 0 , 1 ) . 3 lim n · a n exists and for all x, y in ( 0 , 1 ) , Φ f ( x ) = Φ f ( y ) + lim n · a n ln ( x ( 1 − y ) y ( 1 − x )) . In this case, ω f = ( 0 , 1 ) . Marion Scheepers Rearrangements of numerical series

A detour to groups For x, y in (0,1), define xy x ⊙ y = 1 − x − y + 2 xy . (( 0 , 1 ) , ⊙ ) is an Abelian group with identity element 1 2 . Fact 1: For g a positive real define x Ψ g : ( 0 , 1 ) − → R : x �→ g ln ( 1 − x ) . Ψ g is a group isomorphism from (( 0 , 1 ) , ⊙ ) to ( R , +) . Fact 2: Marion Scheepers Rearrangements of numerical series

Return to Pringsheim’s Theorem C) Let f be signwise monotonic with σ f = ( 0 , 1 ) and Φ f non-constant. Put g = lim n · a n . Then g > 0. Φ f ( · ) − Φ f ( 1 2 ) : ( σ f , ⊙ ) − → ( R , +) is a group isomorphism. The function d ( x , y ) = g | ln ( x ( 1 − y ) y ( 1 − x )) | is a metric on σ f , and measures | � f A − � f B | in terms of d ( A ) and d ( B ) . Marion Scheepers Rearrangements of numerical series