test for convergence or divergence and give rule?

Question

can someone help me out on testing series for convergence or divergence? Please give the rule used too.

1. sum (1 to infinity) of (5^n)n!/(n^n)

2. sum (0 to infinity) of (n^4)/n!

3. sum (1 to infinity) of ((n!)^2) / (2n)!

4. sum (1 to infinity) of ((n-2)/n)^4n

5. sum (1 to infinity) of 4/(n^3 +5)

6. sum (1 to infinity) of 1/(sqrt(n^2 +1))

7. sum (1 to infinity) of 5/(4^n +3)

8. sum (1 to infinity) of (ln n^6) / (3n^5)

Any help on these will be appreciated.

Thanks,

Mark

Answer 1

ratio test = 5(n+1)*n^n/(n+1)(n+1)^n= 5/((1+1/n)^n==>5/e>1
divergent
2)ration test = (1+1/n)^4 /(n+1) ==>0 convergent
3) ratio test = (n+1)^2/(2n+1)*(2n+2) ==>1/4 convergent
4)lim a_n = e^-8 not zero so the series is divergent
5) compare with 1/n^3 The are of the same kind convergent
6) compare with 1/n
7) a_n < 5*(1/4)^n geometric series r= 1/4 convergent
8) as log n

Answer 2

Let me start it.
1. sum (1 to infinity) of (5^n)n!/(n^n)
It diverges by nth root test.