INTRODUCTION

In most calculus courses students learn that the terms of a conditionally convergent series

$$T=\sum_{k=1}^{\infty}a_k(-1)^{k-1}$$ (1)

may be rearranged to converge to any given real number. This result is somewhat mysterious as it seems to contradict our experience. After all the commutativity of addition of real numbers is one of the truth', which we hold as self-evident. Why should this principle break down, when the sum contains an infinite number of terms. Surprisingly, the proof of this result is relatively easy and we will start with a rough sketch of it before shedding some different light on this result. If the sequence $\{a_k\}_{k=1}^{\infty}$ is a decreasing (or at least eventually decreasing) positive sequence that converges to $0$, and $A$ is a positive real number, one obtains a rearrangement that converges to $A$ by first adding odd (positive) terms in the series until

$$\sum_{k=1}^{N-1} a_{2k-1} < A,$$

and

$$\sum_{k=1}^{N} a_{2k-1} \ge A.$$

In the next step one subtracts one or more of the negative terms until the sum is less than $A$. One continues this process by adding positive terms until the sum exceeds $A$ and subtracting negative terms until it is less than $A$ again. In this manner $A$ becomes sandwiched between the partial sums ending with a negative term and the ones ending with a positive term. And since $a_k$ is decreasing to zero, the difference of these partial sums will also go to zero. We refer the reader to [6, pp. 318-319] or [4, p. 518] for a complete proof of this result.

In this paper we will investigate this result from a slightly different angle. For a given rearrangement the $N$-th partial sum

$$S_N(A)$$

contains a unique number $p_N$ of positive terms and $q_N$ of negative terms. In this way the rearrangement can be identified with two sequences of integers

$$\{p_N\}_{N=1}^{\infty},\quad\mbox{and}\quad \{q_N\}_{N=1}^{\infty}.$$

We will first prove a result that connects these sequences in a simple formula to the limit of the rearranged series. This will allow us to compute limits of the series in an efficient way and will lead us to criteria for the convergence of rearrangements. These results are not original, and similar results on the relation between the limits of the series and these two sequences (or related sequences) have appeared in the literature before. A rather complete treatment of these and related problems was given by A. Pringsheim as early 1883 [7]. However, this paper is not easily accessible to most students, as it was written in German and in a rather archaic style. More recent works on this and related subjects are [2,3,8]. Despite the richness of the literature, we felt that the subject deserves further investigation. The proofs of the theorems in this paper are completely elementary and accessible to anyone with a strong background in calculus. In the next section we will state and prove the main result on the relationship between the limits of the series and the sequences $\{p_N\}_{N=1}^{\infty}$ and $\{q_N\}_{N=1}^{\infty}$. The third section of the paper will cover some consequences and simple examples. In the final section we will use the earlier results to prove a limit comparison theorem for rearrangements of series. We will limit ourselves to series with decreasing or eventually decreasing terms.