Show $f(x)=f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o(x-x_0)$ 
Suppose that $f:[a,b]\rightarrow\mathbb{R}$ is $n-1$ times differentiable on $[a,b]$. If $f^{(n)}(x_0)$ exists, then for every $x\in[a,b]$, $$f(x)=f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o(x-x_0)$$ 


Note the when $n=1$, every $x\in[a,b]$ that equals to $f'(x_0)$ where $x_0\in[a,b]$. For $n>1$, we let $$r_n(x)=f(x)-\left(f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n\right)$$ where $r_n(x)$ has $n$ times derivatives, then $r(x_0)=r'(x_0)=\cdots=r_n^{(n)}(x_0)=0$. By the definition of $n$ derivatives, $$r_n^{(n-1)}(x)=r_n^{(n-1)}(x)-r_n^{(n-1)}(x_0)=r_n^{(n-1)}(x_0)(x-x_0)+o(x-x_0)$$ Hence $r_n^{(n-1)}(x)=o(x-x_0)$.

Can someone give me a hint or suggestion to keep going or start a better new proof? I stuck at this step and don't see how to get further. Thanks and Merry Christmas.   
 A: We begin with the identity,
\begin{equation}
f(b) - f(a) = \int_a^b f^\prime(t)dt
\end{equation}
To successively integrate it by parts, introduce functions $\phi_i(t)$, $i \ge 0$ an integer, such that $\phi_0(t) = 1$ and $\phi_i^\prime(t) = \phi_{i - 1}(t)$. We further insist that $\phi_i(b) = 0$ for all $i \ge 1$. We readily see that the functions $\phi_i$ are just the polynomials,
\begin{equation}
\frac{(t - b)^i}{i!}
\end{equation}
We can write the first equation as
\begin{equation}
f(b) - f(a) = \int_a^b \phi_0(t)f^\prime(t)dt = \int_a^b \phi_1^\prime(t)f^\prime(t)dt 
\end{equation}
Integrating by parts,
\begin{eqnarray}
f(b) - f(a) &=& f^\prime(t)\phi_1(t)\Big|_a^b - \int_a^b\phi_1 f^{\prime\prime}(t)dt \\
 &=& f^\prime(b)\phi_1(b) - f^\prime(a)\phi_1(a) - \int_a^b\phi_1 f^{\prime\prime}(t)dt \\
&=& 0 - f^\prime(a)(a - b) - \int_a^b\phi_1 f^{\prime\prime}(t)dt \\
&=& (b - a)f^\prime(a - b) - \int_a^b\phi_1 f^{\prime\prime}(t)dt 
\end{eqnarray}
Integrate the second term above to get,
\begin{equation}
f(b) - f(a) = (b - a)f^\prime(a - b) + \frac{(b - a)^2}{2!}f^{\prime\prime}(a)
+ \int_a^b\phi_2 f^{\prime\prime\prime}(t)dt 
\end{equation}
Repeat the process $n$ times to get the form you want.
