Understanding Maclaurin Series & De Moivre's Theorem

[henryc09]

Homework Statement

[PLAIN]http://img263.imageshack.us/img263/9336/seriesgay.jpg

In the previous part of the question we had to show where the taylor expansion comes from, and calculated the maclaurin series for e^x, sin x and cos x. From that we had to prove De Moivre's theorem and so I would imagine that these things help in the last part of this question. I can see it looks like a Maclaurin series, just not sure where to start really.

Any help would be appreciated, thanks.

 

[Feldoh]

What have you tried? Does that infinite series sort of look like any other infinite series you know?

 

[henryc09]

well it looks like a maclaurin series, but I don't really know how to work out what it's a Maclaurin series of.

 

[Mark44]

Not really. A Maclaurin series has powers of x (or whatever the variable happens to be).

IOW, a Maclaurin series looks like this:
$$\sum_{n = 0}^{\infty} a_n x^n$$

Your series is
$$\sum_{n = 0}^{\infty} \frac{2^n~cos(n\theta)}{n!}$$

 

[henryc09]

Yeah but I was thinking that it looked like something to do with e^(i$$\theta$$) to the power of n which would give terms of cos(n$$\theta$$). I'm not sure, if not how do I go about tackling the problem?

 

[Count Iblis]

Yeah but I was thinking that it looked like something to do with e^(i$$\theta$$) to the power of n which would give terms of cos(n$$\theta$$). I'm not sure, if not how do I go about tackling the problem?

Hint: The real part of exp(i x) = ?

 

[henryc09]

cos(x)... sorry it's been a long day! Still not sure how to use that to get any further.

 

[Count Iblis]

cos(x)... sorry it's been a long day! Still not sure how to use that to get any further.

Replace all the cos(nx) by exp(i n x) in the summation and then take the real part of the summation.

 

[henryc09]

ok, I think I have it.

Is it the Maclaurin series for

e^(2*Re[e^(i$$\theta$$)])

that seems to work I think :s, meaning that the sum is just what's written above right?

 

[henryc09]

or rather:

Re[e^(2*e^(i0))]

 

[Count Iblis]

or rather:

Re[e^(2*e^(i0))]

Yes, and now you can simply this using Euler's formula

exp(ix) = cos(x) + i sin(x)