ggggggjjjj

\begin{equation*}
\begin{aligned}
f(x)=f(0)+\frac{f'(0)}{1!}(x-0)+\frac{f''(0)}{2!}(x-0)^2+\frac{f'''(0)}{3!}(x-0)^3+...\\
=f(0)+f'(0)x+\frac{f''(0)}{2}x^2+\frac{f'''(0)}{6}x^3+\frac{f^{(4)}(0)}{24}x^4\\
+\frac{f^{(5)}(0)}{120}x^5+\frac{f^{(6)}(0)}{720}x^6+\frac{f^{(7)}(0)}{5040}x^7+\frac{f^{(8)}(0)}{40320}x^8\\
=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}
\end{aligned}
\end{equation*}