Which of the following statements are true (root/ratio test)?

A. The Root Test can be used to determine whether the series (from k=1 to infinity) 危 k/(3+k^2) converges or diverges





B. The Ratio Test can be used to determine whether the series (from n=1 to infinity) 危 1/(n^2) converges or diverges.





1) A only


2) both of them


3) neither of them


4) B onlyWhich of the following statements are true (root/ratio test)?
A)





Test the lim n-%26gt;inf |(n/(3+n^2))^(1/n)|





= |e^(lim n-%26gt;inf ln (n/(3+n^2)^(1/n)|


= |e^(lim n-%26gt;inf ln(n/3 + n^2) / n|





That gives infinity over infinity, so now you can use l'hopitals:





= e^(lim n-%26gt;inf (1/3 + 2n) / (n/3 + n^2))


= e^(lim n-%26gt;inf 1 / (n/3 + n^2) + 2 / (1/3 + n))


= e^0


= 1


So this test is indeterminate.





B) For ratio test we will test the value of the following limit:





lim n-%26gt;inf | (1/(n+1)^2) / 1 / n^2)


= lim n-%26gt;inf | n^2 / (n+1)^2 |





Use L'Hopital's:





= lim n-%26gt;inf | 2n / 2(n+1) |


= lim n-%26gt;inf | n / (n+1) |





Use it again:





lim n-%26gt;inf |1 / 1|





So again this test is indeterminate.





So neither of themWhich of the following statements are true (root/ratio test)?
there both stupid.


just take the integral to infinity and if it equals infinity is converges if it equals 0 it diverges.


nuff said
B only
Damian, do the work to show that #3 is correct...just involves some limit taking.