Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$ In complex field, assume $f(x)=\sum_{i=1}^{\infty} a_i x^i$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area.
Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x), $$
or $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),$$
where $p(x)$ is a polynomial with all coefficients which are natural numbers, and $r_i(x)$ is a quotient of polynomials with at least one pole(which means denominator is polynomial with at least one zero point)，$r_i(x)$ can be expanded as $$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{   }b_{ij} \in N\text{ or } b_{ij} = 0 ?$$
Question 2: If they do exist, how to compute them?
Question 3: Under what condition $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),n <\infty?$$
Question 4: Are there finite number of $$T_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{   }b_{ij} \in N\text{ or } b_{ij} = 0 $$,where
$$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{   }b_{ij} \in N\text{ or } b_{ij} = 0 $$,
are all "finitely generated" by $T_i(x)$ by multiply and addition?
 A: Consider the sequence $1,0,0,1,0,0,0,0,1,\dots$ given by $a_n=1$ if $n$ is a square, otherwise $a_n=0$. I claim that no nonzero sequence $b_n$ of non-negative integers with $b_n\le a_n$ for all $n$ can satisfy a constant coefficient homogeneous linear recurrence relation. For any sequence satisfying such a relation of order $d$ with $d$ consecutive terms equal to zero must be the zero sequence, and any sequence of non-negative integers dominated by $a_n$ must have arbitrarily long runs of zeros. 
Now if $r(x)$ is a rational function, then the coefficients of its power series expansion must satisfy a constant coefficient linear recurrence. So if $f(x)=x+x^4+x^9+\cdots$, it can't be written as a sum, finite or infinite, of rational functions with poles and with power series with non-negative integer coefficients. 
EDIT. More generally, a sum of a finite number of rational functions is a rational function, and a power series represents a rational function if and only if its coefficients satisfy a constant coefficient homogenous linear recurrence, so the question of representing $f$ as a finite sum boils down to the question of determining whether the coefficients of $f$ satisfy such a relation. How to do this will depend on how $f$ is known. 
I don't know what to say about infinite sums, other than to give an example. If we write $d(n)$ for the number of divisors of $n$, then $$f(x)=\sum d(n)x^n={x\over1-x}+{x^2\over1-x^2}+{x^3\over1-x^3}+\cdots$$ while $f$ is certainly not a rational function. 
