A series which is not convergent. Series may diverge by marching off to infinity or by oscillating. Divergent series
have some curious properties. For example, rearranging the terms of gives both and .
No less an authority than N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever"
(Gardner 1984, p. 171; Hoffman 1998, p. 218). However, divergent series
can actually be "summed" rigorously by using extensions to the usual
summation rules (e.g., so-called Abel and Cesàro sums). For example, the divergent
series
has both Abel and Cesàro sums of 1/2.