Nested expansions and hardy fields

This short survey of some recent results is based on talks given atthe Universities of Kent and Leipzig. The author would like to take this opportunity to thank Wolfgang Lassner and the University of Leipzig for their excellent hospitality during the last week of November 1992. The use of power series to represent functions and to perform computations involving them is very well established. So also is the fact that natural ways of developing series expansions can often lead to series which diverge. The following example is classical [8]. Let ~(~) = ~~m t-l e~dt, z <0. By successively integrating by parts, we obtain

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