. We saw in Theorem 3.54 that rearranging the terms of a non-absolutely convergent series can change its behavior drastically. But not all rearrangements have such effects. If {a,} is a sequence, consider the rearrangement obtained by interchanging successive pairsa2m and a2m+1 · Clearly, this can be written}, where k, is the sequence of integers{akn2, 1,4, 3, 6, 5, ... , n defined by= K2m,-1 k2m=2m-1 for m =1, 2, ...Show that if an or ak, is a convergent series, then so is the other, andΣαn= Σαkn,

Fig: 1