In my lecture notes it says that you only need to test for absolute convergence of a power series, and that it can be proven that the power series necessarily diverges (even if it is not a positive series) if it fails the ratio test for absolute convergence. I cannot find an explanation of why this would be so.
Conditional convergence of the power series.
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0"you only need to test for absolute convergence of a power series" What do you mean "need"? need for what purpose? What does "fail the ratio test" mean here? – 2017-01-10
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0@zhw. My lecture notes meant that you only need to test for absolute convergence, and even if the series is not positive if it fails the ratio test (which is for positive series and absolute convergence), it diverges even if it is an alternating series. Failing the ratio test means that the limit of the n+1th term and the n the term as n tends to infinity is greater than 1. – 2017-01-10
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Because in the ratio or root test you look at the absolute value of the coefficients. Eg, a power series $\sum_0^\infty a_n z^n$ converges by the ratio test if
$$\limsup \left|\frac{a_{n+1}z^{n+1}}{a_n z^n}\right|=|z|\cdot \limsup\frac{|a_{n+1}|}{|a_n|}<1$$
and diverges if the $\limsup$ is larger than $1$. So the signs of the coefficients $a_n$ are irrelevant to determining convergence or divergence. It's possible for the signs to affect convergence on the radius of convergence.