I working to prove the following statement:
If $\lim \limits_{k \to \infty} \lvert \frac{a_k}{a_{k+1}}\rvert$ exists then the power series $\sum_{k=0}^\infty a_kx^k$ has the convergence radius $$R= \lim \limits_{k \to \infty} \lvert \frac{a_k}{a_{k+1}}\rvert$$
So the proof in my book is making me really confused. I understand that it starts by using the ratio test:
$$y=\lim \limits_{k \to \infty}\lvert\frac{a_{k+1}x^{k+1}}{a_{k}x^k}\rvert = \frac{\rvert x \lvert}{1}\lim \limits_{k \to \infty}\lvert\frac{a_{k+1}}{a_{k}}\rvert = \frac{\lvert x\rvert}{R}$$
I understand that I can use the ratio test in order to see where the series is converging. So the text book is plugging in the power series in the ratio test. I see that $\lvert \frac{x^{k+1}}{x^k}\rvert=\lvert x \rvert$ but what I don't see is how: $$\lim \limits_{k \to \infty}\lvert\frac{a_{k+1}}{a_{k}}\rvert = \frac{1}{R}$$
Then it states that if $\frac{\lvert x \lvert}{R}<1$ the series is absolutely convergent and if $\frac{\lvert x \lvert}{R}>1$ the series is divergent.
Any clues on why the limit equals $\frac{1}{R}$ would be greatly appreciated.