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Math Help - Calc 2 Alternating Series

  1. #1
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    Calc 2 Alternating Series

    1) Classify as absolutely convergent, conditionally convergent, or divergent.

    The sum starting at k=1 to infinity of (sink)/k^3
    2) The series satisfies the hypotheses of the alternating series test. For n = 5, find an upper bound on the absolute error that results if the sum of the series is approximated by the nth partial sum.

    The sum starting at k=1 to infinity of [(-1)^(k+1)]/k!
    3) The series satisfies the hypotheses of the alternating series test. Find a value of n for which the nth partial sum is ensured to approximate the sum of the series to the stated accuracy.

    The sum starting at k=1 to infinity of [(-1)^(k+1)]/[(k+1)ln(k+1)]

    4) Find an upper bound on the absolute error that results if S(sub10) is used to approximate the sum of the given geometric series. Compute S(sub10) rounded to four decimal places and compare this value with the exact sum of the series.

    1-(2/3)+(4/9)-(8/27)+...
    I know this is a lot but any help would be great!
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  2. #2
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    1) Classify as absolutely convergent, conditionally convergent, or divergent.
    The sum starting at k=1 to infinity of (sink)/k^3
    Use the fact that |\sin(x)|\leq 1 for all x.

    2) The series satisfies the hypotheses of the alternating series test. For n = 5, find an upper bound on the absolute error that results if the sum of the series is approximated by the nth partial sum.
    The sum starting at k=1 to infinity of [(-1)^(k+1)]/k!

    3) The series satisfies the hypotheses of the alternating series test. Find a value of n for which the nth partial sum is ensured to approximate the sum of the series to the stated accuracy.
    The sum starting at k=1 to infinity of [(-1)^(k+1)]/[(k+1)ln(k+1)]
    If you have the nth partial sum, then the absolute value of the n+1th term is an upper bound for the error

    4) Find an upper bound on the absolute error that results if S(sub10) is used to approximate the sum of the given geometric series.
    Show that the geometric series satisfies the conditions need for the alternating series test, then use the associated upper bound.

    Compute S(sub10) rounded to four decimal places
    Use the formula for a finite geometric series:
    \sum_{k=0}^nax^k = a\frac{1-x^{n+1}}{1-x}

    and compare this value with the exact sum of the series.
    1-(2/3)+(4/9)-(8/27)+...
    Use the formula for an infinite geometric series:
    \sum_{k=0}^{\infty}ax^k = a\frac{1}{1-x} if |x|<1.
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