The following question was used as an example in my calc textbook:
Find the radius of convergence and interval of convergence of the series:
$\sum_{n=0}^\infty \frac{(-3)^nx^n}{\sqrt{n+1}}$
In the example, they use the ratio test to determine that the series converges when |x|<$\frac{1}{3}$. My question is, doesn't the ratio test for absolute convergence? I understand that absolute convergence implies convergence, but since absolute convergence is stronger than convergence, wouldn't there be values of x in an alternating series where the series is convergent but not absolutely convergent, and therefore this answer only gives part of the solution? Or is that taken into account by testing the end-points of the interval?