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Math Help - Can someone help me justify this step when using the limit comparison test please?

  1. #1
    s3a
    s3a is offline
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    Can someone help me justify this step when using the limit comparison test please?

    So I am given the sum of the expression in the following link: http://www.wolframalpha.com/input/?i=lim+as+n-%3Einf+(+(cos(n))^2+%2F(n^2+%2B+1)+*+n^2+)

    I have a_n = (cos(n))^2 /(n^2 + 1) and b_n = 1/n^2

    on the step, lim n=>inf (cos(n))^2 /(n^2 + 1) * n^2, I don't know what to do with (cos(n))^2 since it fluctuates however, what I did is I just assumed it to be a negligible constant other than 0 (such as 1) and my next step is lim n=>inf n^2/(n^2 +1) = 1 and I then concluded that the answer is convergent (which it is) however I really feel that I just "got lucky" and would appreciate it if someone could explain the (cos(n))^2 part.

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
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    If your series contains trigonometric functions, its not good idea to use the limit comparison test
    just use the basic comparison test

    \frac{cos^2(n)}{n^2+1} \leq \frac{1}{n^2+1} \leq \frac{1}{n^2}

    I think you can complete it
    right ?
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