Power series for $(a+x)^{-1}$ Is it possible to write the following expression in terms of power series?
$$ (a+x)^{-1}=\sum\limits_{k =  - \infty }^\infty  {{b_k}{x^k}} $$
where $0 < a < 1$ and $0 < x < 1$.

You can also use the general binomial series (converging for $|x| < 1$):
$$(1+x)^\alpha = \sum_{k=0}^{\infty}\frac{\alpha(\alpha-1) \ldots (\alpha-k+1)}{k!}x^k.$$
Hence,
$$(a+x)^{-1} = a^{-1}(1 + x/a)^{-1}  = a^{-1}\sum_{k=0}^{\infty}\frac{-1(-2) \ldots (-k)}{k!}\left(\frac{x}{a}\right)^k\\ = a^{-1}\sum_{k=0}^{\infty}(-1)^k\left(\frac{x}{a}\right)^k,$$
converging for $|x| < |a|.$
