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Math Help - Approximating Sum of a Series

  1. #1
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    Approximating Sum of a Series

    Approximating Sum of a Series-imageuploadedbytapatalk-21373743783.808111.jpgwell I know this series converges absolutely after testing by comparison

    Now I need to approximate the sum with an error less than .01

    How do I know how far to go?
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  2. #2
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    Re: Approximating Sum of a Series

    Quote Originally Posted by minneola24 View Post
    Click image for larger version. 

Name:	ImageUploadedByTapatalk 21373743783.808111.jpg 
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ID:	28818well I know this series converges absolutely after testing by comparison.
    First it does not converge absolutely because \sum\limits_n {\frac{{\sqrt n }}{{2n + 1}}} does not converge. WHY?

    But it does converge conditionally using the alternating series test.

    Look-up that test and that will be an approximating rule.
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  3. #3
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    Re: Approximating Sum of a Series

    Since that is an alternating series, you know that the "n+1" sum lies between the "n-1" sum and "n" sum. One of the things that means is that the infinite sum lies between any two consecutive terms. If two terms have a difference of less than .01, either of them is within .01 of the infinite sum.
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    Re: Approximating Sum of a Series

    Ok I did the comparison test and your right I got conditionally convergent.

    But I'm still confused how to calculus the sum with an error less than .01

    Can I do trial and error on this?
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    Re: Approximating Sum of a Series

    Quote Originally Posted by minneola24 View Post
    Ok I did the comparison test and your right I got conditionally convergent.
    But I'm still confused how to calculus the sum with an error less than .01
    Can I do trial and error on this?
    If a_n>0 and \sum\limits_{n = 1}^\infty  {{{\left( { - 1} \right)}^n}{a_n}}  = S and S_n=\sum\limits_{n = 1}^\infty  {{{\left( { - 1} \right)}^n}{a_n}} is a partial sum then \left|S_n-S\right|<a_{n+1}.

    So make \frac{\sqrt{n+1}}{2n+3}<0.01
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    Re: Approximating Sum of a Series

    Thanks that worked perfect and I got the same answer as the book!

    Now a new question how do I deal with factorial a? Such as this question Approximating Sum of a Series-imageuploadedbytapatalk-21373758262.474240.jpg
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    Re: Approximating Sum of a Series

    New questions belong in new threads.

    -Dan
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