Is there a sufficient condition for which derivative of $\sum_{n=0}^{\infty} a_n x^n$ is bounded for all $x \in \mathbb{R}$?

Question

Assume that every power series Bounded for $x \in \mathbb{R}$ , I want to know

if there is a sufficient condition for $(a_n)\neq0$ , for all $n$ so that the derivative of the power series $\sum_{n=0}^{\infty} a_n x^n$ is bounded for all $x \in \mathbb{R}$ ?

Answer 1

Sufficient condition means a condition (on the $a_n$) which implies that there is some constant $c$, such that $\vert\sum_{n=0}^{\infty} a_n x^n\vert\leq c$ for all real $x$.

Any linear combination of functions of the form $x\mapsto\cos(a x+b)$ has the desired property.

So for example the power series defined by the function $\cos(x)+\sin(x)$, $$\sum_{n=0}^\infty a_n x_n, {\rm where~} a_{2m}:=(-1)^m/(2m)!, a_{2m+1}:=(-1)^m/(2m+1)!{\rm~for~ all~}m\in\mathbb{N_0}, $$ is bounded by $c=2$ and all coefficients are different from zero!