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Math Help - Test for Convergence/Divergence (Part II)

  1. #1
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    Test for Convergence/Divergence (Part II)

    Hello All: I have been having a hard time with some homework problems, 5/20 can't be done by me, so I was hoping y'all could help! (Part II)

    #3) Sum (n=1 to Infinity) of tan(1/n)

    Divergence Test doesn't work (lim = 0), so I'm at a loss. Can I use integral test? Is this even a decreasing function? I'm terrible with trig related identities .

    #4) Sum (n=1 to Infinity) of (n!)^n/(n^4n)

    Part of me wants to use ratio test, but would that be n+1!^(n+1) etc. or just n!^(n+1) or neither. Or, I could separate and use root test, but again not sure of the validity of doing that.
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  2. #2
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    Quote Originally Posted by Sprintz View Post
    Hello All: I have been having a hard time with some homework problems, 5/20 can't be done by me, so I was hoping y'all could help! (Part II)

    #3) Sum (n=1 to Infinity) of tan(1/n)

    Divergence Test doesn't work (lim = 0), so I'm at a loss. Can I use integral test? Is this even a decreasing function? I'm terrible with trig related identities .


    This is a positive series and thus we can use the limit test, and as \lim_{n\rightarrow \infty}\frac{\tan \frac{1}{n}}{\frac{1}{n}}=1, the series diverges.


    #4) Sum (n=1 to Infinity) of (n!)^n/(n^4n)

    Part of me wants to use ratio test, but would that be n+1!^(n+1) etc. or just n!^(n+1) or neither. Or, I could separate and use root test, but again not sure of the validity of doing that.

    What about the root test? It seems to me that the behavior of \frac{n!}{n^4} when n\rightarrow \infty\;\; is pretty clear, isn't it?

    Tonio
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  3. #3
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    Yes; thanks for your help. For some reason I thought you couldn't separate n^4n into (n^4)^n, but yeah I guess it is obvious after that.

    Thanks for your help!
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