Convergence of Complex Series: Using de Moivre's Theorem | Homework Help

In summary, the given problem involves using de Moivre's theorem and geometric series to show that the infinite series 1 + W1 + W2 + W3 + ... is convergent for all values of Ө and to find the sum to infinity. The solution involves finding the real part of a geometric series with terms of the form 3^{-n}e^{2n\theta i}. With the use of this approach, the sum is found to be [9 - 3cos2Ө + 3^-N+1cos2(N-1)Ө - 3^-N+2cos2NӨ] / [10 - 6cos2Ө]. Assistance is requested in solving this
  • #1
delefemiaoa
6
0

Homework Statement



Given that Wn = 3-n cos 2nӨ for n = 1, 2, 3, …, use de Moivre’s theorem to show that

1 + W1 + W2 + W3 + … + WN-1 = [ 9 – 3 cos2Ө+ 3-N+1 cos2(N-1)Ө - 3-N+2 cos2NӨ] / [10 – 6cos2Ө]

Hence show that the infinite series
1 + W1 + W2 + W3 + …
is convergent for all values of Ө, and find the sum to infinity

Please i need help on how to solve this above question. though I have posted it before in my blog but was deleted. I don't know the reason for the deletion. I guessed I should have posted it in homework section

Homework Equations





The Attempt at a Solution


I don't have a clue to this question, I have tried to use A.P and G.P formulas but proved difficult. I need help in order to teach my students preparing for external exams in further maths.
 
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  • #2
Wn will be the real part of [tex]3^{-n}e^{2n\theta i}[/tex]. So summing up Wns is the same thing as taking the real part of a geometric series of terms like that.
 
  • #3
I have used ur suggestion,as follows common ration = 3-1 e2Өni, first term = 1, however I got stucked.
 

What is a sum of complex series?

A sum of complex series is the result of adding together an infinite sequence of terms, where each term is a complex number. It is similar to finding the sum of a regular series, but with the added complexity of working with complex numbers.

How do you find the sum of a complex series?

The sum of a complex series can be found by using the formula S = a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio. This formula can be used for both finite and infinite series, as long as the common ratio r is less than 1 in absolute value.

What are some common techniques for evaluating complex series?

Some common techniques for evaluating complex series include using the geometric series formula, using the Binomial Theorem, and using partial fraction decomposition. Additionally, understanding the properties of complex numbers and their operations can also be helpful in solving complex series problems.

What are some real-world applications of complex series?

Complex series have many real-world applications, such as in engineering, physics, and economics. They can be used to model and analyze systems with oscillating behavior, such as electrical circuits and wave phenomena. In economics, complex series can be used to forecast trends in financial data.

What are some common mistakes to avoid when working with complex series?

One common mistake when working with complex series is assuming that the series will always converge to a finite sum. It is important to check for convergence using tools such as the Ratio Test or the Root Test. Additionally, being careful with complex number operations and understanding the properties of complex numbers can help avoid errors when working with complex series.

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