Characterization of positive McLauren series?

Question

A non-negative real polynomial can always be written as $p_1(x)^2+p_2(x)^2$, (or something like this?), where $p_i$ are real polynomials. But can we say something like this for a positive McLauren series? For example they are always equal to $\sum c_ie^{a_ix}$ with a positive sequence $c_i$, and $a_i$ is real?