Expand (12-5x)/(6-5x-x²) in a power series in x, using
Expand $\dfrac{12-5x}{6-5x-x^2}$ in a power series in $x$, using known expansions, and indicate when it is valid.
Explanation of Method
Factoring the denominator and applying partial fractions splits the function into two simple geometric-type terms, each with its own radius of convergence; the combined series is valid only where both hold simultaneously.
- Factor the denominator $6-5x-x^2$.
- Decompose via partial fractions.
- Expand each term as a geometric series and combine.
Step 1 · Factor the denominator
$$ 6-5x-x^2 = -(x^2+5x-6) = -(x-1)(x+6) = (1-x)(6+x). $$
Step 2 · Partial fractions
Write $\dfrac{12-5x}{(1-x)(6+x)}=\dfrac A{1-x}+\dfrac B{6+x}$. Then $12-5x=A(6+x)+B(1-x)$. At $x=1$: $7=7A\implies A=1$. At $x=-6$: $42=7B\implies B=6$. So $$ \frac{12-5x}{6-5x-x^2} = \frac1{1-x}+\frac6{6+x}. $$
Step 3 · Expand each term
$$ \frac1{1-x}=\sum_{n=0}^{\infty}x^n \quad(|x|<1), \qquad \frac6{6+x}=\frac1{1+\frac x6}=\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{6^n} \quad(|x|<6). $$ Combining: $$ \frac{12-5x}{6-5x-x^2} = \sum_{n=0}^{\infty}\left[1+\frac{(-1)^n}{6^n}\right]x^n = 2-\frac56x+\frac{37}{36}x^2-\cdots $$ The combined series converges only where both pieces converge, i.e. $|x|<\min(1,6)=1$.
Summary Table
| Property | Value |
|---|---|
| Series | $\sum\left[1+(-1)^n/6^n\right]x^n$ |
| Poles | $x=1$ and $x=-6$ |
| Region of validity | $|x|<1$ |
Additional Notes
- The radius of convergence, $1$, is set entirely by the nearer pole at $x=1$; the farther pole at $x=-6$ contributes to the coefficient formula but not to the radius, since $\min(1,6)=1$.
- As $n\to\infty$, the term $(-1)^n/6^n\to0$, so the coefficients approach $1$ — the series behaves asymptotically like the simple geometric series $\sum x^n$, with the second partial-fraction term becoming a vanishing correction.
- This closes out the rational-function expansion family alongside Problems 246, 248, 255, and 256: whenever a rational function's denominator factors into distinct linear or low-degree pieces, partial fractions reduce the expansion to a finite combination of geometric series, with the overall radius always set by the closest singularity.