Shouldn't we check for conditionally convergent in ratio test done to see the intervals of convergence in power series?

Question

(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute value of A(n+1)/A(n), infinite series of A(n) converges because it is is absolutely convergent. But what about when series A(n) is conditionally convergent?

Answer 1

The ratio test says that for a series $\sum_{n=1}^\infty a_n$ such that $$\lim_{n\rightarrow\infty} \left|\frac{a_{n+1}}{a_n}\right| = c$$ that if $c<1$ the series is absolutely convergent, and if $c>1$, then the series is divergent.

The test cannot show that a series is conditionally convergent. If a series is conditionally convergent, then either the limit above doesn't exist or $c=1$, and in both cases, we learn nothing from the ratio test.

Summary: If a series is conditionally convergent, then the ratio test will fail to tell you anything about that series.