Showing a (complex) series is (conditionally) convergent.

In summary, the conversation discusses the convergence of the series \sum^\infty_{n=1}1/n \cdot z^n in the open unit ball. It is proven that the series is convergent for all |z| = 1 except for z = 1. In order to prove this, it is necessary to show that the related series \sum^\infty_{n=1}z^n is bounded for |z| = 1 and z ≠ 1. The speaker is able to solve this problem for rational multiples of pi, but needs help for other arguments of z. The conversation ends with a request for assistance in proving that the related series is bounded.
  • #1
shoescreen
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I've been reading a complex analysis book which had an example showing [tex]\sum^\infty_{n=1}1/n \cdot z^n[/tex] is convergent in the open unit ball.

I'm now looking at the case when [tex]|z| = 1[/tex]. Clearly [tex]z = 1[/tex] is the divergent harmonic series, but i know this series is in fact convergent for all other [tex]|z| = 1[/tex].

In order to prove this, i need to be able to show that the related series [tex]\sum^\infty_{n=1}z^n[/tex] is bounded, whenever [tex]|z| = 1[/tex] and [tex]z \neq 1 [/tex],.

I can solve this problem when ever the argument of z is a rational multiple of pi, but other than that I'm stuck. Any help proving that this related series is bounded would be very helpful.

Thanks!
 
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  • #2
If you sum the truncated (at N) series, you have (1-zN+1)/(1-z). For z=1, you have 0/0 (no good). For z≠1, the expression is well defined, so see what happens as N -> ∞.
 

FAQ: Showing a (complex) series is (conditionally) convergent.

1. What does it mean for a series to be convergent?

A convergent series is a series in which the sum of its terms approaches a finite number as the number of terms increases. In other words, the series has a well-defined limit.

2. What is the difference between a conditionally convergent series and an absolutely convergent series?

Absolute convergence means that the sum of the absolute values of the terms in the series is finite. Conditional convergence means that the series is convergent, but not absolutely convergent.

3. How do you show that a series is convergent?

To show that a series is convergent, you need to find its limit. This can be done by using various convergence tests such as the ratio test, the root test, or the integral test.

4. What is a divergence test and when should it be used?

A divergence test is a test used to determine if a series is divergent, meaning it does not have a well-defined limit. This test is useful when other convergence tests are inconclusive or cannot be applied.

5. Can a series be both absolutely and conditionally convergent?

No, a series can only be either absolutely convergent or conditionally convergent. If a series is absolutely convergent, it is also conditionally convergent. However, the opposite is not true.

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