i am confused by the series expansion of $\frac{1}{(a+x)}$ for the general case, where it is not known, wether $|a|<|x|$ or $|x|<|a|$. Can I take from the series expansion wolframalpha just the first two terms of both, i.e. $$ - \frac{1}{a} + \frac{1}{a} - \frac{a}{x^2} - \frac{x}{a^2} + \frac{1}{a^2 x^2} - \frac{x^2}{a^3} $$ Finally, I need an approximation for $\frac{1}{(a+x)}$ where $|a-x|$ small.
Edit: My final goal is to separate to get a completely out of the denominator $1/(a-x)$, i.e. to get an exprassion $$ \sum_n a^{n} f_n(x) $$