  We study two types of series over a real alternative $^*$-algebra $A$. The
first type are series of the form $\sum_{n} (x-y)^{\punto n}a_n$, where $a_n$
and $y$ belong to $A$ and $(x-y)^{\punto n}$ denotes the $n$--th power of $x-y$
w.r.t.\ the usual product obtained by requiring commutativity of the
indeterminate $x$ with the elements of $A$. In the real and in the complex
cases, the sums of power series define, respectively, the real analytic and the
holomorphic functions. In the quaternionic case, a series of this type
produces, in the interior of its set of convergence, a function belonging to
the recently introduced class of slice regular functions. We show that also in
the general setting of an alternative algebra $A$, the sum of a power series is
a slice regular function. We consider also a second type of series, the
spherical series, where the powers are replaced by a different sequence of
slice regular polynomials. It is known that on the quaternions, the set of
convergence of these series is an open set, a property not always valid in the
case of power series. We characterize the sets of convergence of this type of
series for an arbitrary alternative $^*$-algebra $A$. In particular, we prove
that these sets are always open in the quadratic cone of $A$. Moreover, we show
that every slice regular function has a spherical series expansion at every
point.
