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Math Help - Check my work on alternating series problem?

  1. #1
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    Check my work on alternating series problem?

    The original series (-1 is raised to the n+3 power. Couldn't figure out how to raise more than just the bracket with LaTeX):

    \sum_{3}^{inf}(-1)^(n+3)/(7+\sqrt[5]{n+11})

    I took everything except the (-1)^n+3 part, and called it bn. In order for the series to be convergent, bn must:

    1. be decreasing
    2. have a limit of 0 when n->inf


    so, bn = 1/(7+\sqrt[5]{n+11})
    bn+1 is less than bn, so it is obviously decreasing.

    And the limit is obviously 0, because the denominator will grow while the numerator stays 1.

    Am I correct? Also, what does it mean to be "absolute" and "conditionally" convergent? Which one is this?
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  2. #2
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    A series \sum a_n convergent if \sum |a_n| is convergent. This is known as absolute convergence.

    If the property of absolute convergence does not hold, but the series is STILL convergent, then the series is conditionally convergent.
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  3. #3
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    Enclose the whole exponent in braces for the desired effect.

    ( - 1 ) ^ { ( n + 3 ) } gives (-1)^{(n+3)}

    Decreasing and Decreasing to Zero is enough for an alternating series.

    Get rid of the "alternating" (usually absolute values does this) and test it again. Success is "absolute" convergence.
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