A Limit Comparison Theorem for Rearrangements

A LIMIT COMPARISON THEOREM FOR REARRANGEMENTS In the previous section we considered the convergence behavior of some special series. This section is devoted to a more general convergence result. Similar to the rich theory of the convergence of positive series, we will prove a comparison theorem. This will allow us to study the convergence of rearrangements of series with more complicated terms. Theorem 2 Let be two conditionally convergent series, which satisfy the assumptions spelled out in the introduction of this paper. Assume that there is a positive constant such that. Then any rearrangement of will converge if and only if the corresponding rearrangement of converges. Proof: To facilitate the proof of this theorem we need to introduce some notation. Consider a given rearrangement of the alternating series with associated sequence s generality we assume that and of positive and negative terms. Without loss of for sufficiently large denote the -th partial sum of this rearrangement, and denote the -th partial sum of the same rearrangement of . Let Moreover, let and denote the partial sums of the corresponding alternating series. Furthermore, let be two continuous functions with anti derivatives Finally, let Let then there exists an such that and such that for all . Let , then Analogously, we get for . Subtracting the second inequality from the first we get In this step we used the decomposition of into and a positive remainder term, and the fact that . On the right hand side of the last inequality we may interchange and without changing the value of the right hand side. This implies that the same inequality applies to and therefore Next, from the proof of Theorem 1 we have that Now assume that 1 converges, then by Theorem converges to a finite number and hence there exists a such that for all . Next, since sequences and there exists a Hence, for , , and such that all converge, they are Cauchy and therefore it is a Cauchy sequence and it converges. The opposite direction is proved completely analogously. We illuminate the use of Theorem 2 by an example. Consider the series The reader can easily verify that satisfies the conditions of both theorems. Now using l'Hospital's rule we get Therefore, by Corollary 6, any rearrangement of this series converges if and only if Remarks: 1. To best of our knowledge, Theorem 2 is, at least in the form given, a new result, albeit not very useful. A. Pringsheim [7] does a comparison of conditionally convergent series with the alternating harmonic series, by comparing whether is finite or not. This result follows from Theorem 2 by using the alternating harmonic series as one of the two series used in Theorem 2. 2. The condition that is only a sufficient condition for rearrangements of the two series two converge together. As we have seen in Corollaries 2 and 3 rearrangements of, converge together, but the condition of Theorem 2 is obviously not satisfied. 3. The entire paper only applies to series with eventually decreasing terms. There are, however series which do not satisfy this property and which are still conditionally convergent and subject to Riemann's Theorem.