![](/files/happy5.png)
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
.
![$ (a_n) $](/files/tex/0343ff13b16e6d39031bcb59a0d31a300a582fd0.png)
![$ \sum_{n=0}^{\infty} a_n x^n $](/files/tex/722a4892fa6950b75b1122e5f22a3459d58ec674.png)
![$ x \in \mathbb{R} $](/files/tex/e3182ddcb88f01afefb8cee99a8319c4deb28f83.png)
Consider a power series that is convergent for all
, thus defining a function
. Are there criteria to decide whether
is bounded (which e.g. is the case for the series with
for
and
for n odd)? Some general remarks:
- \item A necessary condition for
![$ \sum_n a_n x^n $](/files/tex/97ea5bf4b20cdcf99119b0981fba6f591ea75673.png)
![$ a_0 $](/files/tex/1cea874d38d4a5bbfc282f437aa92e2ac6154844.png)
![$ a_n $](/files/tex/36eeb8d732a363eb2e430c12b322eebf36dc7003.png)
![$ a_n $](/files/tex/36eeb8d732a363eb2e430c12b322eebf36dc7003.png)
![$ a_n $](/files/tex/36eeb8d732a363eb2e430c12b322eebf36dc7003.png)
![$ a_0 $](/files/tex/1cea874d38d4a5bbfc282f437aa92e2ac6154844.png)
![$ a \cos( f(x)) $](/files/tex/2860e1a0233c7d5f139c55a34a6f11d5080c0758.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathbb{R} \to \mathbb{R} $](/files/tex/610e76afa0fcc86727798a74e52cc6d72c4e5f83.png)
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.