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Calculus 2 infinite Series

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine if the following series converges or diverges.

    Ʃ[k=1,inf] tan(k)/(k^2+1)

    2. Relevant equations



    3. The attempt at a solution

    I have no idea how to solve this problem. Now that I think of it, I have never solved a single question about series were I'm asked about convergence or divergence of a series with tangent or cotangent as part of the series. Tangent and cotangent are not defined at multiples of pi/2 excluding multiples of pi, but the series is from k to infinity were k is the set of integers and so the numerator all by itself would never go to positive or negative infinity at any k. Yet I can't seem to come up with a solution to this problem.

    Also just a quick question. If I'm given a particular series in which I know the function which it represents, if the function is undefined at some given points, like for example 1/(2-x) or something of the sort, could I automatically conclude that the series doesn't converge at positive 2 sense the function doesn't?

    Also I have never seen a problem were the interval of convergence included two intervals like [-10,5)(5,22] or something of the sort just [-10,5) if that makes any sense at all. Is it possible to have series were there are two intervals of convergence instead of just one?

    Thank you for any help
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 6, 2011 #2
    =( ah man
     
  4. Nov 7, 2011 #3

    Mark44

    Staff: Mentor

    I posted a message about this series in the Homework Helpers' section. Maybe somebody there will have a good idea. The integral test seems like a way to go, but I haven't carried it all the way through.
     
  5. Nov 7, 2011 #4

    I like Serena

    User Avatar
    Homework Helper

    I believe this series diverges.

    A condition for a series to converge is that it needs to be possible for any epsilon > 0, that there is an N'th term such that all subsequent terms have an absolute value less than epsilon.

    tan(k) can get arbitrarily large as k comes arbitrarily close to pi/2 + m pi for some m.
    However large you want it to be, you can get it for some k.
    So you can make always find a term larger than epsilon.

    However, I'm afraid my argument isn't completely rigorous yet.
     
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