Simplify the following equation [Complex Numbers]

jcurl
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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.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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