Simplify the following equation [Complex Numbers]

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Homework Help Overview

The discussion revolves around simplifying a series involving cosine functions, specifically the expression 1 + cos(θ) + cos(2θ) + cos(3θ) + ... + cos(nθ). The subject area pertains to complex numbers and their relation to differential equations, particularly in the context of Laplace Transforms.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the potential connection between the cosine terms and complex exponentials, noting that cos(nθ) can be expressed as the real part of exp(i*n*θ). There is also consideration of relating the series to a geometric series, although uncertainty remains about isolating the cosine values.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the relationship between cosine functions and complex exponentials, but no consensus has been reached on a specific approach to simplify the series.

Contextual Notes

Participants are navigating the transition into Laplace Transforms, which may influence their understanding of complex numbers and series. There is an indication of uncertainty regarding the initial steps to take in simplifying the given expression.

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Homework Statement


I'm in differential equations right now and we are about to start Laplace Transforms. Our homework is over complex numbers:

Simplify the following equation:
1+cos(\theta)+cos(2\theta)+cos(3\theta)+...+cos(n\theta)


Homework Equations





The Attempt at a Solution


I have no idea where to start. My only guess would be to do this: \sum_{i=0}^n cos(n\theta) but I feel like that's way to easy and not what he is asking for.
 
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If you are going to start Laplace transforms, you likely know about complex numbers. cos(n*theta) is the real part of exp(i*n*theta), yes? Can you relate your question to a geometric series?
 
Dick said:
If you are going to start Laplace transforms, you likely know about complex numbers. cos(n*theta) is the real part of exp(i*n*theta), yes? Can you relate your question to a geometric series?

That's what I was thinking, but I'm not sure how to only get the cos value. Since e^{i\theta} = cos(\theta) + isin(\theta)
 
Hint: scroll up and read Dick's post carefully.
 

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