Convergence of z_n: A Complex Series Question with Alpha Boundaries

lom · 2009-10-13T17:41:21+00:00

z_n=\rho_ne^{i\theta_n}\\ is a series of complex numbers which differs 0 for which -\alpha<=\theta_n<=\alpha\\ A.does \sum z_n\\ converge B.does \sum |z_n|\\ converge

Thread starter lom
Start date Oct 13, 2009
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Convergence

In summary, convergence in a complex series refers to the behavior of the terms in the series as the number of terms approaches infinity. It can be determined in a complex series with alpha boundaries by checking if the series satisfies the criteria of the alpha test. This test involves comparing the series to a related series with known convergence properties. A complex series with alpha boundaries can converge for certain values of alpha and diverge for others, depending on the behavior of the series in relation to the alpha test. The alpha test differs from other convergence tests as it compares the series to a related series and can be used for both real and complex series. However, it may have limitations and may not provide information about the rate of convergence.

Oct 13, 2009

lom

[tex]z_n=\rho_ne^{i\theta_n}\\[/tex] is a series of complex numbers which differs 0 for which

[tex]-\alpha<=\theta_n<=\alpha\\[/tex]

A.does [tex]\sum z_n\\[/tex] converge

B.does [tex]\sum |z_n|\\[/tex] converge

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Oct 13, 2009

Mark44

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What have you tried? Do you know any theorems for determining whether an infinite series converges?

1. What is the definition of convergence in a complex series?

Convergence in a complex series refers to the behavior of the terms in the series as the number of terms approaches infinity. A complex series is said to converge if the terms in the series approach a finite limit as the number of terms increases.

2. How is convergence determined in a complex series with alpha boundaries?

In a complex series with alpha boundaries, convergence is determined by checking if the series satisfies the criteria of the alpha test. This test involves comparing the series to a related series with known convergence properties and using the comparison to determine the convergence or divergence of the original series.

3. Can a complex series with alpha boundaries converge for certain values of alpha and diverge for others?

Yes, it is possible for a complex series with alpha boundaries to converge for some values of alpha and diverge for others. The convergence or divergence of the series is dependent on the specific value of alpha and the behavior of the series in relation to the alpha test.

4. How does the alpha test differ from other convergence tests for complex series?

The alpha test differs from other convergence tests for complex series in that it compares the series to a related series with known convergence properties, rather than directly examining the behavior of the terms in the series. It is also a more general test, as it can be used for both real and complex series.

5. Are there any limitations to using the alpha test for complex series convergence?

While the alpha test is a useful tool for determining convergence in complex series, it does have some limitations. It may not work for all series, and it can be difficult to determine the appropriate related series for comparison. Additionally, it may not provide information about the rate of convergence, which may be important in certain applications.