# Criterion for boundedness of power series

 Importance: Low ✭
 Author(s): Rüdinger, Andreas
 Subject: Analysis
 Keywords: boundedness power series real analysis
 Recomm. for undergrads: yes
 Posted by: andreasruedinger on: May 9th, 2009
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:

• 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.
• Changing a finite set of 's (except ) does leave the subspace of bounded power series.
• 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.

### 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.

### What you have then is a

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

### Re: A necessary condition

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."

### harder than that

Look at sin(x)=x-x^3/6+x^5/120-....

JPB

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

The sine function is in the class mentioned.

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