# Criterion for boundedness of power series

**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.

### What you have then is a

On June 21st, 2012 Anonymous says:

What you have then is a polynomial, and any nonconstant polynomial function is unbounded.

### Re: A necessary condition

On February 16th, 2011 Comet says:

I posted the above comment anonymously, but now I have created an account. "It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n."

### sin x = cos(pi/2 - x)

On June 21st, 2012 Anonymous says:

The sine function is in the class mentioned.

## A necessary condition

It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n.