He Rearrangemens of C1-slummable Series

is absolutely convergent and has the sum s . Then, as is well known, every rearrangement, S' a ;, of (1) also converges and has the same sum s . If, however, (1) n=1 converges, but not absolutely, then, according to Riemann's classical rearrangement theorem [3, p . 235, or 2, p . 318j, for every real number s', there exists a rearrangement of (1) whose sum is s' . Assume, now, that (1) is C 1-summable [1, p . 7, or 2, p. 464], and that its C1sum is a . Consider the set of all C' 1 -summable rearrangements of (1) ; what is the nature of the corresponding set of ('1-sums? We are going to answer this question ; the answer turns out to be somewhat more complicated than Riemann's rearrangement theorem (and also more difficult to obtain). We shall show, namely, that, for any C,-summable series (1), the rearrangement set (cf. Definition 1 below) consists either of a single number, or o f all numbers o f the form a H P I (), 0, + 1, + 2, . . .) for some particular real numbers f3 f0 and a, or of all the real numbers. Moreover, given any x, there exists a C,-summable series (1) whose rearrangement set consists of the single