Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form $$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$ where $c_{2n}\geq 0$ for all $n$.

Is there a sufficient condition on the coefficients $c_{2n}$ under which the sum is bounded for all $z=x\in \mathbb{R}?$

allcoefficients, not just the ones with sufficiently large powers of $x$. $\endgroup$ – Wojowu Dec 20 '16 at 8:52(i)any finite subset of coefficients with positive indices is uniquely determined by all the others, because any perturbation of it changes the series by a polynomial, thus unbounded;(ii)for any $p$, the coefficients $c_{2n}$ corresponding to indices $n$ multiples of $2p+1$ arenotdetermined by all the others, because one may add e.g. the series of $\cos x^{2p+1}$. $\endgroup$ – Pietro Majer Dec 20 '16 at 10:59