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Math Help - Testing for Convergence/Divergence

  1. #1
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    Testing for Convergence/Divergence

    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!

    #1) Sum (n=1 to Infinity) of ((2n+1)^n)/(n^2n)

    I thought I should separate it into ((2n+1)/n)^n) * 1/n, but I wasn't sure whether you can do this. I would us the root test to get find the limit of the first part is 2 > 1 = divergent, but I don't think this makes any sense.

    #2) Sum (n=1 to Infinity) of SQRT(n^2-1)/(n^3+2n^2+5).

    I know this is limit comparison test, and should be easy, but I can't figure out what to use. Surely I should make bsubn = n/n^3 = 1/n, but how do I simplify an/bn?

    #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.

    #5) Sum (n=1 to Infinity) of 1/(n + ncos^2(n))

    I've thought of making the denom. 1(1+cos^2(n)) and using an identity, but nothing simplifies. I could try the Integral test, would this work?

    Thanks all for your help, anything is appreciated!!

    Cheers.
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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!

    #1) Sum (n=1 to Infinity) of ((2n+1)^n)/(n^2n)

    I thought I should separate it into ((2n+1)/n)^n) * 1/n, but I wasn't sure whether you can do this. I would us the root test to get find the limit of the first part is 2 > 1 = divergent, but I don't think this makes any sense.

    #2) Sum (n=1 to Infinity) of SQRT(n^2-1)/(n^3+2n^2+5).

    I know this is limit comparison test, and should be easy, but I can't figure out what to use. Surely I should make bsubn = n/n^3 = 1/n, but how do I simplify an/bn?

    #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.

    #5) Sum (n=1 to Infinity) of 1/(n + ncos^2(n))

    I've thought of making the denom. 1(1+cos^2(n)) and using an identity, but nothing simplifies. I could try the Integral test, would this work?

    Thanks all for your help, anything is appreciated!!

    Cheers.
    Please re-post with no more than 2 questions per thread. Thankyou. Thread closed.
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