Solution
= Solution
The statement
$$
f(x)\sim\sum_{n=0}^{\infty}f_n(x-x_0)^n
$$
as $x\to x_0$ means that for every $N\geq0$,
$$
f(x)-\sum_{n=0}^Nf_n(x-x_0)^n
=o((x-x_0)^N).
$$
= Solution
The statement
$$
f(x)\sim\sum_{n=0}^{\infty}f_n(x-x_0)^n
$$
as $x\to x_0$ means that for every $N\geq0$,
$$
f(x)-\sum_{n=0}^Nf_n(x-x_0)^n
=o((x-x_0)^N).
$$