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Math Help - series tests

  1. #1
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    series tests

    Select the FIRST correct reason on the list why the given series converges.

    a) Geometric series.
    b) Comparison with a convergent p series.
    c) Integral test.
    d) Ratio test.
    e) Alternating series test.

    1) \sum_{n=1}^\infty \frac{1}{n\left(ln\left(n\right)\right)^2}

    2) \sum_{n=1}^\infty \frac{\sin^2 (4 n)}{n^2}

    3)  \sum_{n=1}^\infty\frac{n^2+\sqrt{n}}{n^4 -3}
    im struggling a bit on series section, how do you know what the best test for each problem?
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  2. #2
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    1)Use the integral test with \int_2^{\infty} \frac{dx}{x \ln^2 x } (I think your index such start with 2).

    2)This is a compasion test \left| \frac{\sin ^2 (4n)}{n^2 } \right| \leq \frac{1}{n^2} and \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} < \infty.

    3)Another comparasion test \left| \frac{n^2 + \sqrt{n}}{n^4 - 3} \right| \leq \frac{n^2+n^2}{n^4 - \frac{1}{2}n^4} = \frac{4n^2}{n^4} = \frac{2}{n^2}.

    This is Mine 63th Post!!!
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  3. #3
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    thank you ThePerfectHacker and congratulation
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