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Infinite Series

  1. Dec 8, 2003 #1
    Hi all!
    I was wondering which method one should use to find the actual sum of an infinite series. I know how to find the sum of a geometric series (if it converges), but how could I find the sum for, for instance

    I know that it converges, and the sum appears to be 2. But how can I calculate this?

    Or how about
    \sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}

    Thanks in advance,
  2. jcsd
  3. Dec 8, 2003 #2


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    There is no general method for determining the sum of an infinite series.
  4. Dec 8, 2003 #3
    What if we try to find the sum to n terms and then taking lim [tex]n\rightarrow\infty[/tex]
  5. Dec 9, 2003 #4
    The answer to the first one is 2.5746952396343726343 Hope that will help
    Last edited: Dec 9, 2003
  6. Dec 9, 2003 #5
    The second one is a defined convergent series:

    [tex]\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}=\frac12\left(1+\pi{\rm csch}(\pi)\right)[/tex]

    where csch(z) gives the hyperbolic cosecant of z, or in other words: csch(z)=1/sinh(z).
    Last edited: Dec 9, 2003
  7. Dec 9, 2003 #6
    How You guys reach this conclusions I have read a little about convergence and divergence but don't know how you summed up the series
  8. Dec 13, 2003 #7
    This one does not involve hyperbolic trig functions.

    Taylor series.
  9. Dec 13, 2003 #8


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    laura: would you mind explaining further? Since the sum, as written, is clearly not a Taylor series, do you mean that it can be converted to one and then summed? If so, how? It's certainly not obvious to me!
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