Given the power series , prove that if exists and is finite or infinite, then it is equal to the radius of convergence of the series.

Suggestion : Use the ratio test.

My attempt : The ratio test states that if exists and if then the series converges. If then we can't conclude by the test. If then the series diverges.

The radius of convergence of the power series is .

Thus I must show that .

I first assume that , in other words that .

I've tried to continue the proof ( by showing using the fact that ) but without success.

Can you help me a little bit? Thanks.