\input amstex

$$\underset {x\rightarrow a}\to {f(x)}=\sum\limits_{k=1}^n\frac{f^{(k)}(a)}{k!}(x-a)^k+\underset{x\rightarrow a}\to {o\bigl((x-a)^n\bigr)}$$

$$\aligned
[a\equiv0,\ f\equiv\ln(1+x)]\;\Longrightarrow\;\underset {x\rightarrow 0}\to {\ln(1+x)}&=\sum\limits_{k=1}^n\frac{\bigl(\ln(1+x)\bigr)^{(k)}\big|_{x=0}}{k!}\cdot x^k+\underset{x\rightarrow 0}\to {o(x^n)}=\\
&=\sum\limits_{k=1}^n\frac{(-1)^{k-1}\cdot(k-1)!}{(1+x)^k}\bigg|_{x=0}\cdot\frac{1}{k!}\cdot x^k+\underset{x\rightarrow 0}\to {o(x^n)}=\\
&=x-x^2/2+x^3/3-x^4/4+\dotsc+\underset {x\rightarrow0}\to{o(x^n)}
\endaligned$$
\bye