First order convergence explanation understanding

Question

I was wondering if i am understanding the reasoning wrong or if the source has a mistake?

Source:

To my understanding it should read: $$\frac{x_2-x_1}{x_1-x_0} = \frac{k^2\varepsilon-k\varepsilon}{k\varepsilon-\varepsilon}=\frac{k\varepsilon(k-1)}{\varepsilon(k-1)}=k$$ Does $x_2 - x_1 = k\varepsilon-k^2\varepsilon$ or does $x_2 - x_1 = k^2\varepsilon-k\varepsilon$ if so why?

it looks like the difference is multiplying the top and the bottom by an extra factor of $-1$, which cancels out, and hence there is no practical difference. — gt6989b, Jun 19 '17 at 19:46
Is there a reason for this? is it considered better formatting to have $k^2\varepsilon - k\varepsilon$ as the $k^2$ is in front and its a higher power? — Sonny Da Silva-Peters, Jun 19 '17 at 19:48
I updated my answer to explain which is typically denoted better formatting — infinitylord, Jun 19 '17 at 20:06

score 2 · Accepted Answer · answered Jun 19 '17 at 19:59

You are correct and their algebra is wrong. As they put it: $$\frac{k \epsilon-k \epsilon^2}{\epsilon - k \epsilon} = \frac{k \epsilon (1 - \epsilon)}{\epsilon (1-k)} = \frac{k(1 - \epsilon)}{1-k} \ne k$$

As for your question in the comments, I find it better style to put $$\frac{k \epsilon - k^2 \epsilon}{\epsilon - k \epsilon} \space \text{ instead of } \space \frac{k^2 \epsilon - k \epsilon}{k \epsilon - \epsilon}$$ For the sole reason that in the first notation, both the numerator and denominator are positive, and in the second, they are both negative (as $0 < k < 1$)

First order convergence explanation understanding

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