Extracting the coefficient of $x^n$ from a fraction

Question

I need help extracting the coefficient of $x^n$ from a $\frac{1-x}{1-2x}$.

So far I have that

\begin{align} \frac{1-x}{1-2x} &= \frac{1}{1-2x} - x\frac{1}{1-2x}\\ &= \sum\limits_{k=0}(2x)^k - x\sum\limits_{k=0}(2x)^k \end{align}

would it then be correct to say that

\begin{align} [x^n]\frac{1-x}{1-2x} = 2^n - 2^{n-1}? \end{align}

Answer 1

Indeed.

Why ? For the two series you consider are of positive radius, hence you can identify the coefficients, as in any equality between integer series of positive radius.

And for your 'correct' answer, it's the same : $2^n - 2^{n-1} = 2^{n-1}(2-1)$... so, that's it.