\input amstex

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

$$\aligned
[a\equiv 0,\ f\equiv\sin]\;\Longrightarrow\;\underset
 {x\rightarrow0}\to {\sin(x)}&=\sum\limits_{k=1}^n\frac{\sin^{(k)}(0)}{k!}x^k+\underset
 {x\rightarrow0}\to {o(x^n)}\\
&=\cos(0)\cdot x/1-\sin(0)\cdot x^2/2-\cos(0)\cdot x^3/6+\sin(0)\cdot x^4/24+\cos(0)\cdot x^5/120+\dotsc+\underset {x\rightarrow0}\to {o(x^n)}=\\
&=x-x^3/6+x^5/120+\dotsc+(-1)^{n-1}\cdot x^{2n-1}/(2n-1)!+\underset {x\rightarrow0}\to {o(x^{2n-1})}
\endaligned$$
\bye