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Math Help - Absolute convergence

  1. #1
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    Absolute convergence

    Match each of the following with the correct statement.
    A. The series is absolutely convergent.
    C. The series converges, but is not absolutely convergent.
    D. The series diverges.

    1.
    2.
    3.
    4.
    5.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by amiv4 View Post
    Match each of the following with the correct statement.
    A. The series is absolutely convergent.
    C. The series converges, but is not absolutely convergent.
    D. The series diverges.

    1.
    2.
    3.
    4.
    5.

    1.

    \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{n-1}}{2^{n+1}n^{\frac{1}{5}}}=\frac{1}{4}\sum_{n=1}^  {\infty}\frac{(-1)^n}{n^{\frac{1}{5}}}

    This series converges by Leibniz's Criterion since \frac{1}{(n+1)^{\frac{1}{5}}}\leq\frac{1}{n^{\frac  {1}{5}}} and \frac{1}{n^{\frac{1}{5}}\to{0}. But the absolute value diverges by the integral test.

    2. Eliminate the common factors and compare with \frac{1}{3^n}
    3. We see that

    \sum_{n=0}^{\infty}\left|\frac{(-1)^n}{5^nn!}\right|=\sum_{n=0}^{\infty}\frac{1}{5^  nn!} which converges by any number of tests, most handily the Root or Ratio. And therefore since the absolute value of the kernel converges so does the original series, thus absolute convergent.

    4. Diverges by the n-th term test

    5. Converges by limit comparison test with \frac{1}{5^n}
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