1. ## 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! (Split up due to rules)

#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).

2. Originally Posted by Sprintz
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! (Split up due to rules)

#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).
#1) Use the nth root test. The nth root is (2n+1)/n^2 --> 0.

#2) Use the comparison test, noting that SQRT(n^2-1)/(n^3+2n^2+5) < n/n^3 = 1/n^2.