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Math Help - series diverging

  1. #1
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    series diverging

    I am having trouble understand divergence and convergence. I am trying to solve this problem, Show that each of the following series diverges.
    3 + 5/2 + 7/3 + 9/4

    Do i need to find a S(n) formula then take the limit of it as n goes to infinity? I know to use limits, but I don't understand what the answer to the limit tells you. Thanks for the help!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by davecs77 View Post
    I am having trouble understand divergence and convergence. I am trying to solve this problem, Show that each of the following series diverges.
    3 + 5/2 + 7/3 + 9/4

    Do i need to find a S(n) formula then take the limit of it as n goes to infinity? I know to use limits, but I don't understand what the answer to the limit tells you. Thanks for the help!
    Let S_n = 3 + \frac {5}{2} + \frac {7}{3} + \frac {9}{4} + ...

    then S_n = \sum_{n = 1}^{ \infty} \frac {2n + 1}{n}

    note that \lim_{n \to \infty} S_n = 2 \neq 0

    thus the series diverges by the test for divergence


    here is what the answer to the limit tells you. if the answer is zero then the series MIGHT converge, if it isn't zero, then it definitely does not converge


    Theorem: Let S_n be a sequence. If \sum S_n converges, then \lim_{n \to \infty} S_n = 0

    Note, this is an implication, the converse is not always true
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    Let S_n = 3 + \frac {5}{2} + \frac {7}{3} + \frac {9}{4} + ...

    then S_n = \sum_{n = 1}^{ \infty} \frac {2n + 1}{n}

    note that \lim_{n \to \infty} S_n = 2 \neq 0

    thus the series diverges by the test for divergence


    here is what the answer to the limit tells you. if the answer is zero then the series MIGHT converge, if it isn't zero, then it definitely does not converge


    Theorem: Let S_n be a sequence. If \sum S_n converges, then \lim_{n \to \infty} S_n = 0

    Note, this is an implication, the converse is not always true

    How would you know if the series converges if the limit is 0? Can you do a test for that?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by davecs77 View Post
    How would you know if the series converges if the limit is 0? Can you do a test for that?
    if the limit was zero, we would need another test. there are many to choose from: the comparison test, the integral test, the limit comparison test, ...
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    Quote Originally Posted by davecs77 View Post
    How would you know if the series converges if the limit is 0? Can you do a test for that?
    For example,
    \sum_{n=1}^{\infty}\frac{1}{n} diverges even though \lim \frac{1}{n}=0.
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