Criterion for boundedness of power series

Importance: Low ✭

Author(s):

Rüdinger, Andreas

Subject:

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 $(a_n)$ so that the power series $\sum_{n=0}^{\infty} a_n x^n$ is bounded for all $x \in \mathbb{R}$ .

Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all $x \in {\mathbb R}$ , thus defining a function $f: {\mathbb R} \to {\mathbb R}$ . Are there criteria to decide whether $f$ is bounded (which e.g. is the case for the series with $a_n = (-1)^k/(2k)!$ for $n = 2k$ and $a_n = 0$ for n odd)? Some general remarks:

$\sum_n a_n x^n$

$a_0$

$a_n$

$a_n$

$a_n$

$a_0$

$a \cos( f(x))$

$f$

$\mathbb{R} \to \mathbb{R}$

add new comment

A necessary condition

On February 10th, 2011 Anonymous says:

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

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."

harder than that

On April 27th, 2012 Anonymous says:

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

JPB

sin x = cos(pi/2 - x)

On June 21st, 2012 Anonymous says:

The sine function is in the class mentioned.