Importance: Low ✭
 Author(s): Rüdinger, Andreas
 Subject: Analysis
 Keywords: boundedness power series real analysis
Question   Give a necessary and sufficient criterion for the sequence so that the power series is bounded for all .
Consider a power series that is convergent for all , thus defining a function . Are there criteria to decide whether is bounded (which e.g. is the case for the series with for and for n odd)? Some general remarks:
\item A necessary condition for to be bounded is that is the only non-zero or there are infinitely many non-zero 's which change sign infinitely many times. \item Changing a finite set of 's (except ) does leave the subspace of bounded power series. \item The subspace of bounded power series is "large" in the sense that it is both a linear subspace (closed under sums and scalar multiples) and an algebra (closed under products). It includes all functions of the form , where is any entire function . The question whether the subspace of bounded power series contains only these functions seems to be open.