Error estimates of power series $ \left| f(x) - \sum_{n=0}^k a_n(x-x_0)^n \right| \leq \left(\frac{r}{R_1} \right)^{k+1} \frac{R_1}{R_1-r} $ 
Suppose that $f(x)=\sum a_n(x-x_0)^n$ has a radius of convergence $R$ and $0<r<R_1<R$. Show that there is an integer $k_0$ such that $$ \left| f(x) - \sum_{n=0}^k a_n(x-x_0)^n \right| \leq \left(\frac{r}{R_1} \right)^{k+1} \frac{R_1}{R_1-r} $$
  if $|x-x_0| \leq r$ and $k \geq k_0$. 

I have to admit I am a bit lost to approach this problem. What is the best way to proceed? 
If this problem is impossible, How would you prove that no integer $k$ can exist in this configuration?
any input/help is much appreciated.

Note that the radius of convergence is $R$ and $R_1 <R$. Thus $f$ is absolutely convergent when $|x-x_0|\le R_1$. In particular, when $|x-x_0| = R_1$, 
$$\sum_{n=1}^\infty |a_n| R_1^n <\infty \Rightarrow |a_n | R_1^n \to 0.$$
Thus there is $k_0$ so that $|a_n| \le R_1^{-n}$ for all $n\ge k_0$. Now for all $r<R_1$ and $|x-x_0|\le r$, if $k \ge k_0$, 
$$\begin{split}
\left| f(x) - \sum_{n=0}^k a_n (x-x_0)^n\right| &= \left| \sum_{n=k+1}a_n (x-x_0)^n\right|\\
&\le \sum_{n=k+1}^\infty |a_n| |x-x_0|^n \\
&\le \sum_{n=k+1}^\infty R_1^{-n} r^n \\
&= \left(\frac{r}{R_1}\right)^{k+1} \sum_{n=0}^\infty \left( \frac{r}{R_1}\right)^n \\
&=\left(\frac{r}{R_1}\right)^{k+1} \frac{1}{1-(r/R_1)} \\
&=\left(\frac{r}{R_1}\right)^{k+1} \frac{R_1}{R_1 -r}. 
\end{split}$$
