Conditional convergence of the power series.

Question

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.

Answer 1

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.