Quote by HallsofIvy
What's wrong with using the comparison test is that it only applies to positive series. Certainly n< (1/2)^{n} for all n, but we can't use that to conclude that [itex]\sum n [/itex] converges!
The crucial point is that every a_{n} is positive. That means that [itex]\sum a_n x^n[/itex], for x negative is an alternating series. What is true of alternating series?

All of terms a_n*R^n are positive and it's convergent. The series is absolutely convergent. Nothing can go wrong here.