Complex Power Series: Does Absolute Convergence Imply Convergence?

[Niles]

Homework Statement

Hi all.

In my book on complex analysis, they discuss complex power series. They use a variety of "tests" to determine absolute convergence, but they never say if this also implies convergence.

Does it?


Niles.

 

[HallsofIvy]

If a series convergerges "absolutely" then, yes, it must converge. The other way is not true: if a seriies converges, it does not necessariily converge absolutely. For example, the series
$$\sum_{n=0}^\infty \frac{(-1)^n}{m}$$
converges but does not converge absolutely.

Most of the time, when you are dealing with complex series, you are dealing with power series. In that case, there always exist a "radius of convergence". Inside that radius, the power series muist converge absolutely, outside it, diverge. But on the radius of convergence, the series may converge absolutely, converge but not absolutely (converge "conditionallly"), or diverge.

 

[Niles]

Thank you very much.