  Puiseux series are power series in which the exponents can be fractional
and/or negative rational numbers. Several computer algebra systems have one or
more built-in or loadable functions for computing truncated Puiseux series --
perhaps generalized to allow coefficients containing functions of the series
variable that are dominated by any power of that variable, such as logarithms
and nested logarithms of the series variable. Some computer-algebra systems
also offer functions that can compute more-general truncated recursive
hierarchical series. However, for all of these kinds of truncated series there
are important implementation details that haven't been addressed before in the
published literature and in current implementations.
  For implementers this article contains ideas for designing more convenient,
correct, and efficient implementations or improving existing ones. For users,
this article is a warning about some of these limitations. Many of the ideas in
this article have been implemented in the computer-algebra within the TI-Nspire
calculator, Windows and Macintosh products.
