Expand f(x)=dfrac1a-x (a ≠0) in 1^circ powers of x
Expand $f(x)=\dfrac1{a-x}$ ($a\ne0$) in $1^{\circ}$ powers of $x$, $2^{\circ}$ powers of $(x-b)$ with $b\ne a$, $3^{\circ}$ powers of $1/x$. Determine the region of convergence in each case.
Explanation of Method
All three expansions come from the same geometric series $\frac1{1-t}=\sum t^n$, applied to three different rewritings of $\frac1{a-x}$ that isolate a small quantity $t$: $x/a$, $(x-b)/(a-b)$, and $a/x$ respectively. The region of convergence is always $|t|<1$, translated back into the corresponding variable.
- Factor $a-x$ to expose $x/a$, then apply the geometric series.
- Shift to center $b$ via $y=x-b$, exposing $y/(a-b)$.
- Factor $a-x$ around $-x$ instead, exposing $a/x$, valid for large $|x|$.
Step 1 · Powers of $x$
$$ \frac1{a-x} = \frac1a\cdot\frac1{1-x/a} = \frac1a\sum_{n=0}^{\infty}\left(\frac xa\right)^n = \sum_{n=0}^{\infty}\frac{x^n}{a^{n+1}}, \qquad |x|<|a|. $$
Step 2 · Powers of $(x-b)$
With $y=x-b$: $a-x=(a-b)-y$, so $$ \frac1{a-x} = \frac1{a-b}\cdot\frac1{1-\frac y{a-b}} = \sum_{n=0}^{\infty}\frac{(x-b)^n}{(a-b)^{n+1}}, \qquad |x-b|<|a-b|. $$
Step 3 · Powers of $1/x$
For large $|x|$, factor out $-x$ instead: $$ \frac1{a-x} = \frac{-1}{x\left(1-\frac ax\right)} = -\frac1x\sum_{n=0}^{\infty}\left(\frac ax\right)^n = -\sum_{n=1}^{\infty}\frac{a^{n-1}}{x^n}, \qquad |x|>|a|. $$
Summary Table
| Expansion | Region of convergence |
|---|---|
| Powers of $x$ | $|x|<|a|$ |
| Powers of $(x-b)$ | $|x-b|<|a-b|$ |
| Powers of $1/x$ | $|x|>|a|$ |
Additional Notes
- Parts $1^{\circ}$ and $3^{\circ}$ are complementary: their regions $|x|<|a|$ and $|x|>|a|$ partition the entire plane except for the single singularity at $x=a$ — the two series are the two halves of the full Laurent expansion of $1/(a-x)$ centered at $x=0$.
- Part $2^{\circ}$ generalizes Part $1^{\circ}$ (which is the special case $b=0$): the region of convergence is always the largest disk centered at the expansion point that avoids the pole $x=a$, exactly a radius $|a-b|$.
- This three-way expansion pattern — near the origin, near an arbitrary point, and near infinity — is the standard template used throughout complex analysis for expanding a meromorphic function around any point or region avoiding its poles, and every subsequent expansion problem in this batch (247–260) is a variation on the same geometric-series idea.