Let's say I have a power series like $\sum_{n=0}^{\infty}a_n x^{b_n}$ For example, with $a_n=\frac{(-1)^n}{(2n+1)!}$ and $b_n=2n+1$, it's the sine function so it's bounded.
If I have to choose $a_n$ and $b_n$, what conditions should I have on them to be sure the power series will be bounded (at least in its convergence interval) ? (I don't even know if it's possible to find such conditions, but my final goal would be to get a non-periodic bounded power series or polynomial, so if you have another idea, I'll take it)