Radius of convergence problem

Question

I need a hint on this problem, been staring on the blackbord a long time now. Problem: Suppose $f(x) = \sum a_{n}(x-x_{0})^n$ has radius of convergence $R$ and $0 < r < R_{1} < R$. Show that there exist an integer k such that $$|f(x) - \sum_{n=0}^{k} a_{n}(x-x_{0})^n| \leq (\frac{r}{R_{1}})^{k+1}\frac{R_{1}}{R_{1}-r}$$ if $|x-x_{0}| \leq r$ and $n > k$.

Answer 1

$$\left|f(x)-\sum_{n=0}^ka_n(x-x_0)^n\right|=\left|\sum_{n=k+1}^\infty a_n(x-x_0)^n\right|\le\sum_{n=k+1}^\infty |a_n||x-x_0|^n\le$$

$$\le\sum_{n=k+1}^\infty \left(\frac r {R_1}\right)^n\stackrel{\text{Geom. Series}}=\left(\frac r{R_1}\right)^{k+1}\frac1{1-\frac r{R_1}}$$

You want now to prove the second inequality (the rightmost one in the first line)...

Further hint: Cauchy-Hadamard Formula.