Rearrangements of Series

Definition: Letbe an infinite series. Ifis any one to one function fromontothen the infinite seriesis called a rearrangement of

With absolutely convergent series all rearrangements converge to the same limit:

Proof: For eachletandconverges toChoose then there issuch that forsince converges absolutely.is a one to one mappingontohence there is K>=N such thatAssumethen

Thusconverges to

Conversely ifis an infinite series with the property that every rearrangement converges to the same limit, the series is absolutely convergent. Ifconverges conditionally then the sum of all the positive termsdiverges and the sum of all the negative terms diverges.

Theorem

Ifconverges conditionally then given any real number r there is a rearrangement ofthat converges to r.

Proof: Supposeandare unbounded sequences andconverges to zero. Letbe the first positive integer such thatThere is such ansinceis an unbounded increasing sequence. Next letbe the first positive integer such thatLetbe the least positive integer such thatContinuing in this way we obtain a rearrangement converging tosinceconverges to zero.