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Infinite series

  1. Nov 13, 2012 #1
    Question says: [itex]\sum(cos(n*pi)/5^n)[/itex] from 0 to infinity.

    Proved that it converges: ratio test goes to abs(cos(pi*(n+1))/5cos(pi*n)) with some basic algebra. As n goes to infinity, this approaches -1/5 (absolute value giving 1/5) since cos(pi*(n+1))/cos(pi*n) is always -1, excepting the asymptotes.

    Question wants to find sum. Wolfram claims sum is 5/6 and won't elaborate. How?
     
  2. jcsd
  3. Nov 13, 2012 #2
    Edit: Nevermind, misread what you wrote. My bad.
     
    Last edited: Nov 13, 2012
  4. Nov 13, 2012 #3
    The ratio test, I believe, is to use the limit as n goes to infinity of a(n+1) / a(n). So for detail:

    cos (pi*(n+1))/5^(n+1) * 5^(n)/cos(pi*n).

    I believe 5^n reduces with 5^(n+1) in the denominator, leaving 5 in the denominator, does it not?
     
  5. Nov 13, 2012 #4

    Dick

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    Write out the first few terms of your series. You have a geometric series in disguise. That's how WA is summing it.
     
  6. Nov 13, 2012 #5
    Thank you.
     
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