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Math Help - Convergence of the alternate harmonic series

  1. #1
    Junior Member RaisinBread's Avatar
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    Convergence of the alternate harmonic series

    Greetings,

    I have been assigned to prove the theorem in the pdf attachment. Convergence of the alternate harmonic series-theorem.pdf

    I found the general idea behind the proof of the theorem in our textbook, so I know what I have to show and I see why it works, but I'm having a little trouble rigorously writing out one part of it.

    I am trying to show that the sums of (Xpo)n and (Xne)n diverge to plus and minus infinity respectively.

    So far, I have explored 2 alternative solutions, but both seems to lead me to a dead end.

    Using (Xpo)n as an example for my first solution, I figured all I need to do is show that for every real number N, there exists a j such that the sum from 1 to j on the sequence (Xpo)n is greater than N. Now it seems to me that the only way I can prove that is by finding a way to express the sum from 1 to j of (Xpo)n as a function of j, such that we see at which j the series becomes greater than N, but I am unable to express the sum as a function of j, in fact I'm not even sure this is possible...

    My second solution would be that, we could show that the sums of (Xpo)n and (Xne)n diverge using some convergence criteria, and that since they are both monotone, the sum of (Xpo)n is always increasing and the sum of (Xne)n is always decreasing, It seems to me that they have to be going towards infinity, otherwise we would get contradiction, but I can't quite figure out what is the contradiction.

    So I was wondering if one of you guys had an Idea or a hint to give me about this proof, or if there is something really obvious that I didn't see. Also feel free to tell my if you need any clarifications about my attempted solutions if there is anything I wrote that you don't understand.
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  2. #2
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    Quote Originally Posted by RaisinBread View Post
    Greetings,

    I have been assigned to prove the theorem in the pdf attachment. Click image for larger version. 

Name:	Theorem.pdf 
Views:	33 
Size:	35.9 KB 
ID:	21199

    I found the general idea behind the proof of the theorem in our textbook, so I know what I have to show and I see why it works, but I'm having a little trouble rigorously writing out one part of it.

    I am trying to show that the sums of (Xpo)n and (Xne)n diverge to plus and minus infinity respectively.

    So far, I have explored 2 alternative solutions, but both seems to lead me to a dead end.

    Using (Xpo)n as an example for my first solution, I figured all I need to do is show that for every real number N, there exists a j such that the sum from 1 to j on the sequence (Xpo)n is greater than N. Now it seems to me that the only way I can prove that is by finding a way to express the sum from 1 to j of (Xpo)n as a function of j, such that we see at which j the series becomes greater than N, but I am unable to express the sum as a function of j, in fact I'm not even sure this is possible...

    My second solution would be that, we could show that the sums of (Xpo)n and (Xne)n diverge using some convergence criteria, and that since they are both monotone, the sum of (Xpo)n is always increasing and the sum of (Xne)n is always decreasing, It seems to me that they have to be going towards infinity, otherwise we would get contradiction, but I can't quite figure out what is the contradiction.

    So I was wondering if one of you guys had an Idea or a hint to give me about this proof, or if there is something really obvious that I didn't see. Also feel free to tell my if you need any clarifications about my attempted solutions if there is anything I wrote that you don't understand.

    I think you want the following, a really ashtonishing result: Riemann series theorem - Wikipedia, the free encyclopedia

    Tonio
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  3. #3
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    Quote Originally Posted by RaisinBread View Post
    I am trying to show that the sums of (Xpo)n and (Xne)n diverge to plus and minus infinity respectively.
    The series of even-numbered terms is -\frac12-\frac14-\frac16-\ldots = -\frac12\bigl(1+\frac12+\frac13+\ldots\bigr), which diverges to -\infty because of the well-known fact that the harmonic series diverges.

    For the series of odd-numbered terms, notice that each odd-numbered term is greater in absolute value than the following even-numbered term. Then use the comparison test to deduce that the series of odd-numbered terms also diverges (to +\infty).
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  4. #4
    Junior Member RaisinBread's Avatar
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    Interesting, I never suspected that Wikipedia would have such elaborate mathematical proofs on it.

    @Opalg Thank you very much! The solution is so simple, can't believe I didn't see that!
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