Complex Power Series: Does Absolute Convergence Imply Convergence?

In summary, the conversation discusses the use of tests in determining absolute convergence of complex power series. It is stated that if a series converges absolutely, it must also converge. However, the opposite is not always true. The concept of a radius of convergence is also mentioned, where inside the radius the series must converge absolutely, outside it diverges, and on the radius it may converge absolutely, converge conditionally, or diverge.
  • #1
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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.
 
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  • #2
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
[tex]\sum_{n=0}^\infty \frac{(-1)^n}{m}[/tex]
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.
 
  • #3
Thank you very much.
 

1. What is a complex power series?

A complex power series is a mathematical series that involves complex numbers. It is written in the form of ∑(an(z-c)n), where an is a complex number, z is the variable, and c is a constant. It is used to represent complex functions and can be used to approximate other functions.

2. What is absolute convergence?

Absolute convergence refers to the convergence of a series when the absolute value of its terms approach zero. This means that the series will converge regardless of the order in which the terms are added. It is a stronger form of convergence compared to conditional convergence.

3. Does absolute convergence imply convergence for complex power series?

Yes, absolute convergence does imply convergence for complex power series. This is because if a series is absolutely convergent, it automatically satisfies the conditions for convergence. Therefore, if a complex power series is absolutely convergent, it will also be convergent.

4. Are there any cases where absolute convergence does not imply convergence for complex power series?

No, there are no cases where absolute convergence does not imply convergence for complex power series. This is a fundamental property of complex power series and holds true for all cases.

5. How is absolute convergence determined for complex power series?

To determine if a complex power series is absolutely convergent, the ratio test or the root test can be used. If the limit of the ratio or the root is less than 1, the series is absolutely convergent. If it is greater than 1, the series is divergent. If it is equal to 1, the test is inconclusive and another method must be used.

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