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Math Help - Convergent Subsequences....

  1. #1
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    Convergent Subsequences....

    I am reviewing my analysis midterm to study for the final, and I can't seem to remember how to get the correct solution to one of the problems.


    It says: consider the sequence {xn} with
    {xn}=((n+1)^2+(-1)^n*n^2)/(n*(n+1))

    Determine if it converges or diverges.

    I got that it diverges, which is correct, but I didn't get credit because my method was invalid

    I just did a simple divide through by the highest term and you see that it boils down to 1+(-1)^n, which diverges. Of course you can't actually do this because in doing so I took terms like 1/n^2 and said they converge to 0, and so, this is what you are left with.

    The professor said the correct solution involves finding two subsequences which converge to different values. I am not sure how to do this. Any help please? Thanks.
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  2. #2
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    We can agree that x_n  = \frac{{\left( {1 + \frac{1}{n}} \right)^2  + \left( { - 1} \right)^n }}{{1 + \frac{1}{n}}}.

    y_n  = x_{2n}  = \frac{{\left( {1 + \frac{1}{{2n}}} \right)^2  + 1}}<br />
{{1 + \frac{1}{{2n}}}} \to 2.

    z_n  = x_{2n - 1}  = \frac{{\left( {1 + \frac{1}{{2n - 1}}} \right)^2  - 1}}{{1 + \frac{1}{{2n}}}} \to 0
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  3. #3
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    Oh I see...you just took the even terms and the odd terms. Makes sense, thanks.
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