  We investigate power series that converge to a bounded function on the real
line. First, we establish relations between coefficients of a power series and
boundedness of the resulting function; in particular, we show that boundedness
can be prevented by certain Tur\'an inequalities and, in the case of real
coefficients, by certain sign patterns. Second, we show that the set of bounded
power series naturally supports three topologies and that these topologies are
inequivalent and incomplete. In each case, we determine the topological
completion. Third, we study the algebra of bounded power series, revealing the
key role of the backward shift operator.
