Convergence/divergence of complex series

In summary, the question is about proving the divergence of the series \sum(-i)n/ln(n) from 2 to infinity. While it may seem that the series is not absolutely convergent, this is not enough to prove its divergence. However, by using the Leibniz theorem, it can be shown that the real and imaginary parts of the series form two alternating series that converge to a finite number. This contradicts the claim that the series is divergent and suggests that the answer provided may be incorrect. A second opinion should be considered.
  • #1
freddyfish
57
0
The question is simple:

how do I prove that the following series is divergent? That it is not absolutely convergent is not hard to see, but that is not enough to prove divergence of the series as it is presented:

[itex]\sum[/itex](-i)n/ln(n)

The summation is from 2 to infinity.

Thanks
 
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  • #2
freddyfish said:
The question is simple:

how do I prove that the following series is divergent? That it is not absolutely convergent is not hard to see, but that is not enough to prove divergence of the series as it is presented:

[itex]\sum[/itex](-i)n/ln(n)

The summation is from 2 to infinity.

Thanks

I don't think it is divergent. The real and imaginary parts form two alternating series, don't they?
 
  • #3
Yeah, they sure do and by the Leibniz theorem they converge to some finite number. The book that the answer is taken from stinks in the sense that the key is very often wrong, so I guess this is one of the cases where it is wrong and now I also have a second opinion to take into consideration.

Thank you for your answer! :)
 

FAQ: Convergence/divergence of complex series

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

The convergence/divergence of a complex series is a measure of whether the series approaches a finite value (converges) or does not approach a finite value (diverges) as the number of terms in the series increases.

2. How do you determine if a complex series is convergent or divergent?

There are several tests that can be used to determine the convergence or divergence of a complex series, including the ratio test, the root test, and the comparison test. These tests involve analyzing the behavior of the terms in the series and comparing them to known convergent or divergent series.

3. What is the significance of a series being convergent or divergent?

The convergence or divergence of a series is important in many areas of mathematics and science, as it can help determine the behavior of a function or the validity of a mathematical model. In addition, the convergence or divergence of a series can also provide insights into the underlying properties of the numbers or variables involved in the series.

4. Can a complex series have both convergent and divergent parts?

Yes, it is possible for a complex series to have both convergent and divergent parts. This is known as a conditionally convergent series, where the series as a whole is divergent, but certain rearrangements of the terms can result in a convergent series. This concept can be counterintuitive and requires careful analysis of the series to determine its behavior.

5. Are there real-world applications of convergence/divergence of complex series?

Yes, the concept of convergence and divergence of complex series has many real-world applications in fields such as physics, engineering, and economics. For example, the convergence of a power series can be used to approximate functions and solve differential equations, while the divergence of a series can indicate the limits of a mathematical model. Additionally, the convergence or divergence of a series can also be used to analyze the behavior of systems and predict outcomes.

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