Formal summation of divergent series (Q1270683)

The author gives the technique of ``telescoping'' a series. The curious and well-known feature of the technique is that it provides a ``sum'' for any oscillatory and divergent series as well as for many convergent series. The author shows that telescoping can be put on a rigorous foundation, and consistency with other methods of summabilities are guaranteed. At the end the author defines telescopic extensions for any summation method, and he proves some Tauberian theorems and other related theorems for telescopic extensions of some classical summation methods. The results proved are interesting and new.