Sums of entire surjective functions Suppose $(f_n)_n$ is a countable  family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero,  such that $\sum_{n=1}^{\infty} a_n f_n$ is entire but not-surjective? In fact, I am interested in this question under the additional assumption that $(f_n)_n$ are not polynomials. 
 A: One expects there to be no such $a_n$ in general, because the
"typical" entire functions is surjective (those that aren't are of the 
special form $z \mapsto c + \exp g(z)$).  An explicit example is
$f_n(z) = \cos z/n$: any convergent linear combination $f = \sum_n a_n f_n$ 
is of order $1$, so if $f$ is not surjective then $g$ is a polynomial
of degree at most $1$; but $f$ is even, so must be constant,
from which it soon follows that $a_n=0$ for every $n$.
A: The answer is no. If something does not hold for polynomials, don't expect that it will hold for entire functions:-)
For example, all non-constant functions of order less than $1/2$ are surjective.
This follows from an old theorem of Wiman that for such function $f$ there exists
a sequence $r_k\to\infty$ such that $\min_{|z|=r_k}|f(z)|\to\infty$ as $k\to \infty.$
And of course linear combinations of functions of order less than $1/2$ are of order less
than $1/2$.
Edit. To construct a counterexample with infinite sums, one can use lacunary series. Let $\Lambda$ be a sequence of integers $n_k$ which grows sufficiently fast,
for example, such that $n_k/k\to\infty$,
and consider the class of entire functions of the form
$$f(z)=\sum_{n\in\Lambda}c_nz^n.$$
It is known that all such functions are surjective. And of course any linear combination of such functions, finite or infinite, belongs to the class.
Reference: L. Sons, An analogue of a theorem of W.H.J. Fuchs on gap series,
Proc. LMS, 1970, 21 525-539.
