  The article is devoted to the alternating Cantor series.
  It is proved that any real number belonging to $[a_0-1;a_0]$, where
$a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more
than two representations by the series and only numbers from countable subset
of real numbers have two representations. Geometry of the representation,
properties of cylinders sets and semicylinders, simplest metric problems are
investigated. Some applications of the series in fractal theory and relation
between positive and alternating Cantor series are described. A shift operator
and its some applications, set of incomplete sums are studied. Necessary and
sufficient conditions for representations of rational numbers by the
alternating Cantor series are formulated.
