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Math Help - test for convergence

  1. #1
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    test for convergence

    show that the sum of 1-cos(pie/n) from n = 1 to infinity converges or diverges



    second question:
    show that the sum of e^-n/(n+1) from n =1 to infinity converges or diverges
    i think it converges. is it okay if i use the comparison test by comparing series to another series and can i use the integral test to see if series being compared converges?
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  2. #2
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    Doesn't the second one diverge by Nth term/TFD or am I mistaken?
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  3. #3
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    can someone please clearly show/explain answers to the two questions?
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  4. #4
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    #1: We're going to make use of the fact that for all x \in (0, \pi), we have: 0 < 1 - \cos x < \tfrac{1}{2}x^2. To prove this, consider f(x) = 1 - \cos x - \tfrac{1}{2}x^2 and prove that it is a decreasing function. Thus, f(x) < f(0) = 0 \ \Leftrightarrow \ 1 - \cos x < \tfrac{1}{2}x^2

    So now we have the inequality: 0 < 1 - \cos \tfrac{\pi}{n} < \frac{\pi^2}{2n^2} since \frac{1}{n} \in (0, \pi) for all n.

    Now use comparison.

    #2: Use the divergence test. If \lim_{n \to \infty} a_n is not equal to 0, then the series \sum a_n diverges.
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  5. #5
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    thanks

    thanks
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  6. #6
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    how'd u get 1/2x^2
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  7. #7
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    It's simply a fact: 0 < 1 - \cos x < \frac{x^2}{2} for 0 < x < \pi
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