# Thread: Test for Convergence/Divergence (Part II)

1. ## 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.

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! (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 $\displaystyle \lim_{n\rightarrow \infty}\frac{\tan \frac{1}{n}}{\frac{1}{n}}=1$, 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 $\displaystyle \frac{n!}{n^4}$ when $\displaystyle n\rightarrow \infty\;\;$ is pretty clear, isn't it?

Tonio

3. 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.