Formal Summation of Divergent Series

The idea of telescoping a series is widely known, but is not widely trusted. It is often treated as a formalism with no meaning, unless convergence is already established. It is shown here that even for divergent series, the results of telescoping are self-consistent, and consistent with other well-behaved summation operations. Moreover, the summation operations obtained by telescoping are the strongest possible operations with these properties. Some Tauberian theorems are exhibited for telescoping.