divergent series

A divergent series is one whose unending addition never settles on a finite value. The running total might march off to infinity, plunge to minus infinity, or swing back and forth forever without ever homing in — in each case there is no single number that the partial sums approach.

Formally, sum a_n diverges precisely when its sequence of partial sums (s_N) fails to have a finite limit. This umbrella covers several distinct behaviours: divergence to +infinity (the partial sums of 1 + 1 + 1 + ...), divergence to -infinity, and oscillatory divergence with no limit at all (the partial sums of 1 - 1 + 1 - 1 + ... alternate between 1 and 0). All are called divergent, though their pictures differ sharply.

Sometimes a divergent series can still be assigned a value by a summation method (Cesàro, Abel, and others) that averages or smooths the partial sums; under Cesàro summation, 1 - 1 + 1 - 1 + ... is assigned 1/2. Such assignments do not contradict divergence in the ordinary sense — they are deliberately weaker notions of summation, useful in their own right but not the same as the limit of partial sums.