The definition of function series Regarding the definition of a "function series" 

$$\sum_{n=0}^{\infty} f_n = f_0+f_1+...$$

such as:

$$\sum_{n=0}^{\infty} \frac{nx}{1+n^4x^2}$$

Does the function series term-wise (as a sequence of terms) look like

$$(0, \frac{1x}{1+1^4x^2}, \frac{2x}{1+2^4x^2}, ...)$$

then?
Be aware of the difference between a map $f$ and the value $f(x)$ of $f$ at some point $x$; knowing this, you know that the series 
$$
\sum_{n=0}^{\infty} \frac{nx}{1+n^4x^2}
$$
is the series whose general term is the value $f_{n}(x)$ of the map $f_{n}: \xi \mapsto \frac{n\xi}{1+n^{4}\xi^{2}}$ at some point $x$.
Yes, the terms of your function series would be the $f_i$, just as you list them. But presumably you are really interested in the sequence of partial sums:

$$\sum_{0 \le k \le n} \frac{k x}{1 + k^4 x^2}$$