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Prove sum identities

  1. May 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Prove the following identities:
    [tex]\sum_{n=0}^{\infty} r^n \cos(n\theta) = \frac{1-r\cos(\theta)}{1-2r\cos(\theta)+r^2} [/tex]

    [tex]\sum_{n=0}^{\infty} r^n \sin(n\theta) = \frac{r\sin(\theta)}{1-2r\cos(\theta)+r^2} [/tex]

    2. Relevant equations
    [tex]\cos(n\theta) = \frac{1}{2}(e^(in\theta)+e^(-in\theta))[/tex]


    3. The attempt at a solution
    I get it to this point...

    [tex] \frac{1}{2}( \sum_{n=0}^{\infty}(re^(i\theta))^n+\sum_{n=0}^{\infty}(re^-(i\theta))^n[/tex]

    But I dont know what to do next! r could be less than or greater than one. I need to do the general case where r could be either....

    Also, Im not sure that the 'relevent equation' is really relevent

    Any help/tips would be greatly appreciated, Thx!
     
    Last edited: May 1, 2007
  2. jcsd
  3. May 1, 2007 #2

    Dick

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    You are now looking at a pair of geometric series. The next step should be easy. I wouldn't worry about the case r>1. In general it should diverge except for special angles.
     
  4. May 3, 2007 #3
    Wouldn't you just use the following rule for an infinite geometric series:

    [tex]\sum\limits_{k = 1}^\infty {ar^{k} = \frac{a}{{1 - r}}}[/tex]
     
    Last edited: May 3, 2007
  5. May 3, 2007 #4
    Yes, but only if R<1. The problem conspicuously leaves out that stipulation, which is where my confusion set it.

    I went ahead and solved it assuming R<1 and ignoring the other case.... I hope thats good enough for some credit.
     
  6. May 3, 2007 #5
    Did you take [tex]r=re^{i\theta}[/tex] and [tex]a=1[/tex]? Now you've gotten me wanting to solve this problem, lol...
     
  7. May 3, 2007 #6
    I took [tex] re^{i\theta} = r(cos\theta + isin\theta) [/tex] and a=1. I have it completed assuming r<1. We will look at later.
     
  8. May 4, 2007 #7

    Dick

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    If r=1 then the series is going to be, at best, only conditionally convergent (r>1, not even that except for some silly special cases like theta=pi). I would really only worry about r inside of the radius of convergence of the series. You can't 'prove the identities' if the sum of the series doesn't exist.
     
    Last edited: May 4, 2007
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