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Math Help - Test for absolute convergence

  1. #1
    Junior Member
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    Test for absolute convergence

    hello. i have to test for convergence and discuss if its absolte convergent or not:

    \sum \frac{(-1)^n} {\sqrt {n}}

    I think i should use the Leibniz Rule, but not sure how to apply it here...
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by coobe View Post
    hello. i have to test for convergence and discuss if its absolte convergent or not:

    \sum \frac{(-1)^n} {\sqrt {n}}

    I think i should use the Leibniz Rule, but not sure how to apply it here...
    an integral test on f(x) = \frac{1}{\sqrt{x}} should do it.
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  3. #3
    Senior Member Twig's Avatar
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    Hi

    \displaystyle\sum_{k=1}^{\infty} |\frac{(-1)^{k}}{\sqrt{k}}| =  \displaystyle\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}

    And on this series you could use the integral test on  h(x) = \frac{1}{\sqrt{x}}

    And this series is divergent because the integral is divergent.

    But \displaystyle\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k}} is convergent by Leibniz convergene theorem.

    Because the series is alternating, decreasing and  \lim_{n\to \infty} a_{n} = 0
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  4. #4
    MHF Contributor matheagle's Avatar
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    The absoute series is a p-series.
    You should know when a p-series converges or diverges.
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