For the following series ∑∞an determine if they are convergent or divergent. If convergent find the sum.
(ii) ∑∞n=0 cos(θ)2n+sin(θ)2n[/B]
geometric series, [/B]
The Attempt at a Solution
First I have to show that the equation is convergent.
Both cos(θ) and sin(θ) alternate between 1 and -1 depending on the value of θ. It was hinted to me by my instructor that I am to define a θ that the series is convergent.
Because cos(θ)2n is basically the same as as r2n, the same rule follows that it will be convergent if |r|<1. I need to find a θ that will let the cos and sin functions fit this rule. [/B]
Cos(0) = 1 and sin (pi/2) = 1. If i give the range 0<θ<(pi/2) then both terms in the series will be convergent because their magnitudes will be less than 1.
"find the sum"
here is where I am having trouble. I have a rule that Σ∞rn = 1/(1-r) which does not seem to apply when you have r2n
Another problem I am having is that I cannot seem to define the sum in terms of θ. I believe that according to the question, the only appropriate way to find a sum is in terms of θ, but i do not have the tools to do this.
What would be the right way to attempt this problem? Right now in my course we are learning about ways of figuring out convergence or divergence of series and have actually done very little on the subject of actually finding the sum of these series. I have done some research and haven't found anything on my issue with r2n. Additionally if anyone could recommend me a good resource for learning this material, could you link it to me? My instructor doesn't use a book and seems to be diving into the deep end very quickly.
I greatly appreciate any help you guys can give me.