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Math Help - Sequence Question (sorry not sure if this is right category)

  1. #1
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    Sequence Question (sorry not sure if this is right category)

    Test whether the sequence converges or diverges.

    A(subscript n)=((-1)^(n-1) * n)/(n^+ 1)

    I was completely lost in this question, but i attempted anyway

    I was able to determine you need to divide numerator and denominator by n isolating (-1)^n-1 on top. I also brought the -1^-1 to the front to simplify things, however, my book didn't do that.

    I know you needed to use the squeeze theorem but the (-1)^n threw me off thogh and when i checked my solutions manual (which is poorly laid out for the record) they use the absolute value of A(subscript n) in their squeeze theorem

    Using the squeeze theorem they let 0/n = 0< |a(sub n)| = 1/(n+1/n) < 1/n

    I understand how this can make sense as the limits of the outer equations both equal 0, making the limit of the inner equation equal 0, but why is it legal to just use 1 (i'm assuming 1 to equal |1^n| in this case) instead of (-1)^n?

    Thanks in advance!
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  2. #2
    MHF Contributor Calculus26's Avatar
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    solution

    For any alternating sequence lim (-1)^n *an

    converges to 0 if liman = 0 and diverges otherwise

    since limn/(n+1) =1 the sequence diverges if you will the subsequences of even and odd terms converge to 1 and -1

    For animations and discussion of this theorem see the following link on my website

    Infinite Sequences
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