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Math Help - Absolute value convergence vs. convergence (Are they the same?)

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    Absolute value convergence vs. convergence (Are they the same?)

    The question is if the absolute value of a sequence converges then does the sequence itself converge. My gut reaction to this is no because you could have something with opposite signs (+ - + -) that would converge in absolute value but diverge otherwise, or even converge to different values. I know that lim n-> infinity |n/-n|=1 and n/-n =-1 but they both converges. No luck on the convergent, divergent scenario. (Keep in mind this is just some random idea about this problem from my head). What do you think?
    Last edited by CountingPenguins; June 27th 2011 at 11:52 PM. Reason: More thought about this.
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    Re: Absolute value convergence vs. convergence (Are they the same?)

    That's the idea indeed. Maybe you could take an explicit example u_n:=(-1)^n.
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    Re: Absolute value convergence vs. convergence (Are they the same?)

    Thanks. I know that putting |1/(-1)^n| converges to 1 and without the absolute value it doesn't. Thanks again.
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    Re: Absolute value convergence vs. convergence (Are they the same?)

    Quote Originally Posted by CountingPenguins View Post
    The question is if the absolute value of a sequence converges then does the sequence itself converge. My gut reaction to this is no because you could have something with opposite signs (+ - + -) that would converge in absolute value but diverge otherwise, or even converge to different values. I know that lim n-> infinity |n/-n|=1 and n/-n =-1 but they both converges. No luck on the convergent, divergent scenario. (Keep in mind this is just some random idea about this problem from my head). What do you think?
    A series is said to be convergent if you can show that it satisfies a convergence test. Testing for absolute convergence is merely one of those tests. In other words, if a function is absolutely convergent, it is convergent, however, if a function is convergent it may or may not be absolutely convergent.

    A perfect example is the alternating harmonic series. It is convergent and has a value of \displaystyle \ln{2}, but it is not absolutely convergent, because the series of absolute values is the harmonic series, which is divergent.
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