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

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#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.