fa.functional analysis - Criteria for boundedness of power series

Consider a power series $sum_{n=0}^{infty} a_n x^n$ that is convergent for all real
x, thus defining a function $f: mathbb{R} to mathbb{R}$.
Can one give necessary and sufficient criteria the sequence of the coefficients $(a_n)$ has to meet in order for $f$ to be bounded on $mathbb{R}$? (Let's disregard the trivial case that $a_0$ is the only non-zero coefficient and let's call a sequence "function-bounded" if the power series is bounded.) Criteria for boundedness seem to be far more difficult to obtain than the usual criteria for convergence of a power series, here some remarks:

a) A necessary condition for $sum_n a_n x^n$ to be bounded is that there are infinitely many non-zero coefficients which change sign infinitely many times.

b) The boundedness of $f$ is an "unstable" property of the sequence of coefficients: any non-zero change in any finite subset (except $a_0$) will destroy boundedness. Thus the linear subspace of all function-bounded sequences is rather "sparse" in the vector space of all sequences representing convergent power series.

c) On the other hand, the linear subspace of all function-bounded sequences contains at least all power series of functions that can be written as $cos circ h$ with $h$ an entire, real-analytic function, and the algebraic span of these functions. One could conjecture that this space is already the space of all bounded functions that can be represented as power series[EDIT: seems to be refuted, cf. comment below]. And perhaps this could be a starting point for deducing the criteria.

EDIT (conjecture added):
Is is true, that every power series $sum_{n=0}^{infty} a_n x^n$ that is convergent for all real $x$ can be modified only by changing the signs of the terms to a convergent power series $sum_{n=0}^{infty} epsilon_n a_n x^n, quad epsilon_n in {pm1}$ that is bounded for all real $x$?
Example: One can modifify the signs of the power series of the exponential function $sum_{n=0}^{infty} x^n/n!$ pretty easily to a bounded power series by $epsilon_n = +1$ for $n = 0 or 1 mod 4$ and $epsilon_n = -1$ for $n = 2 or 3 mod 4$, yielding the function $sin(x) + cos(x)$. (One can modify the signs pretty easily a bit further such that the power series is not only bounded on the real axis, but also on the imaginary axis - but this is not the question here).
I have neither succeeded in finding a counterexample nor in prooving this conjecture.

EDIT2:
Thanks for the nice counterexample. I would like to improve the conjecture as follows: Define a power series $sum_{n=0}^{infty} a_n x^n$ as nondominant, if for all $x in mathbb{R}$ the absolute value of every term $a_kx^k$ is smaller or equal than the sum of the absolute values of all the other terms: $|a_kx^k| le sum_{n neq k} |a_n x^n|$. The improved conjecture is: Is is true, that every nondominant power series $sum_{n=0}^{infty} a_n x^n$ that is convergent for all real $x$ can be modified only by changing the signs of the terms to a convergent power series $sum_{n=0}^{infty} epsilon_n a_n x^n, quad epsilon_n in {pm1}$ that is bounded for all real $x$?