Taylor Polynomial Variable Question

Question

When you have a polynomial that you set your function equal to in the taylor polynomial (centered around $x = a$) $$function = c_0+c_1(x-a)+c_2 (x-a)^2+...$$ why is your variable $(x-a)$. Oddly enough, I used to know why but now I can't remember... happens to me all the time. Thanks!

Answer 1

You want to expand in powers of a small parameter (so that the further you go in the series, the less significant the terms become). If $x$ is a small deviation from $a$ then it looks like $x=a+\epsilon$, with $\epsilon=x-a$ small.