(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?
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$\begingroup$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.
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