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. 



Sums of entire surjective functions
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 $n$.


