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    Integral convergence or divergence

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    The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not started infinite series.

    In the problem attached below, there are 3 integrals. Please note that there are two questions to be answered. 1)To discuss the convergence or divergence of each integral and 2)To discuss whether or not the integrals are absolutely convergent. The problem also asks to give reasons to each answer.

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    Solution Preview

    Please see the attached file.

    First, the p-integral test:

    The integral is convergent if and only if p>1

    Then, the comparison test:

    If f(x) and g(x) are functions defined on [a,b] such that for any then we have:

    If is convergent, so is
    If is divergent, so is

    For the first integral we note that at ...

    Solution Summary

    This shows how to discuss convergence (and absolute convergence) and divergence of integrals. Absolute convergent values are discussed.