A two-sided series, also called a bilateral series, is a series whose indices range over all integers. The series 
converges when
its two tails converge separately, in which case its sum
is
 |
Equivalently, the partial sums 
have a common finite limit
as 
and 
tend independently to infinity.
The symmetric partial sums instead require 
. They can converge through cancellation even when
the two tails diverge. For example, take 
and 
for 
. Every symmetric partial
sum is zero, but the two tails are the harmonic
series and its negative. The symmetric limit is a discrete
Cauchy principal value.
