Approximating sums of even cosine powers with fractional exponents

[icystrike]

Homework Statement

Show that $$cos^{2}(\theta)+cos^{4}(\theta)+cos^{6}(\theta)+cos^{8}(\theta)\approx 4cos^{4.3128}(\theta) , \mid\theta\mid\leq\pi/2$$

and

$$cos^{2}(\theta)+cos^{4}+...+cos^{30}(\theta)\approx 15cos^{11.38211}(\theta) , \mid\theta\mid\leq\pi/2$$

Homework Equations

The Attempt at a Solution

Any method?

 

[LCKurtz]

Well, I don't see how to prove it. Can you give us the context where this problem came from? Where did you find it?

 

[Berko]

1/(1-x) = 1+x+x^2+...

Isn't there a similar formula if the series terminates?

 

[icystrike]

LCKurtz: I've actually came up with this question myself cos i saw this relationship for any sum of even power cosine while doing a problem ...

Berko: What do u exactly mean? Please enlighten me :)

 

[LCKurtz]

1/(1-x) = 1+x+x^2+...

Isn't there a similar formula if the series terminates?

Sure, the problem is a geometric series. And

$$\sum_{k=1}^4 \cos^{2k}(\theta) = \frac{\cos^2(\theta)-\cos^{10}(\theta)}{1-\cos^2(\theta)}$$

How does that help?

 

[icystrike]

Yes! I do recognise that this is a geometric series but nevertheless this cannot lead us to further simplification...

 

[Berko]

hmm yah I simplified somewhat but not sure where those exponents come from.