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.
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.
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.
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.
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.
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.