Some Theorems on Subseries

1. Absolutely convergent series. A simple calculation reveals that the arithmetic mean value of all subsums (including the void sum) of a given finite sum sn=ui+U2 + • • • +un is equal to sn/2. In this section we shall show (see Theorem 1 below) that an integral mean value can be found, consistent with the preceding, for the sums of all infinite subseries of a given absolutely convergent series ^Uk = s. We begin by defining a one-to-one correspondence between the set of all infinite subseries of a given absolutely convergent real series ^Uk = st and the set of all points on the interval 7 = (0 < £ ^ 1). If £ is any point of I then £ admits a unique nonterminating binary representation of the form