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Math Help - Converging/ Diverging Series

  1. #1
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    Converging/ Diverging Series

    Determine whether the following series converge absolutely, converge conditionally, or diverge.

    ∑(n=1,∞) (-1)^n/n√n

    ∑(n=1,∞) [(-1)^(n-1)][n/n(^2)+1]

    My textbook has pretty sparse information in this chapter and I'm really having some trouble understanding the examples. What is the difference between series that converge conditionally and absolutely?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by schnek View Post
    Determine whether the following series converge absolutely, converge conditionally, or diverge.

    ∑(n=1,∞) (-1)^n/n√n

    ∑(n=1,∞) [(-1)^(n-1)][n/n(^2)+1]

    My textbook has pretty sparse information in this chapter and I'm really having some trouble understanding the examples. What is the difference between series that converge conditionally and absolutely?
    \sum_{n=1}^{\infty}\frac{(-1)^n}{n\sqrt{n}} = \sum_{n=1}^{\infty}\frac{(-1)^n}{n^{3/2}} Taking the absolute value of the summand gives you a p-series with p=3/2. What does this tell you?

    \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{n^2+1} Use the alternating series test. As for whether it converges absolutely, compare it to \sum_{n=1}^{\infty}\frac{1}{n}.
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by schnek View Post
    Determine whether the following series converge absolutely, converge conditionally, or diverge.

    ∑(n=1,∞) (-1)^n/n√n

    ∑(n=1,∞) [(-1)^(n-1)][n/n(^2)+1]

    My textbook has pretty sparse information in this chapter and I'm really having some trouble understanding the examples. What is the difference between series that converge conditionally and absolutely?
    Let \sum_{n=1}^{\infty}a_{n} represent an arbitrary infinite series. If \sum_{n=1}^{\infty}|a_{n}| converges, then the orginal series \sum_{n=1}^{\infty}a_{n} converges absolutely. However, if the series \sum_{n=1}^{\infty}a_{n} converges, but \sum_{n=1}^{\infty}|a_{n}| does not converges, then the orginal series \sum_{n=1}^{\infty}a_{n} converges conditionally.

    I was too slow! Hope it helps anyways.
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