Series Comparison Test, help?

[ani9890]

Series Comparison Test, URGENT help?

Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter I (for "incorrect") if any part of the argument is flawed.

1. For all n>2, ∑ 1/ (n^2−1) < 1/n^2 so converges
2. For all n> 1, ∑ arctan(n) / n^3 < pi / 2n^3 so converges
3. For all n>1, ∑ ln(n)/n^2 < 1/n^1.5 so converges
4. For all n>1, ∑ 1/nln(n) < 2/n so diverges
5. For all n>2, ∑ n/(n^3 - 8) < 2/n^2 so converges
6. For all n>2, ∑ ln(n)/n > 1/n so diverges

I believe (1) is incorrect.
(2), (3), and (6) I believe are correct.
But I can't figure out 4 and 5 ? I think 4 is incorrect but I'm not sure, and for 5 shouldn't it be compared to 1/n^2 ?

Please help!

 

[Bohrok]

What's your reasoning behind the numbers you have answers for?

 

[ani9890]

for (1) 1/n^2 is actually greater than 1/ (n^2−1), for (4) if it diverges it should actually be asubn > bsubn not the other way around, and (5) the bsubn should be n/n^3 which becomes 1/n^2

for the (2),(3) 1/ n^p where p>1 so it converges which is correct, and for (6) n^p, here p=1 so it diverges which is correct.

 

[Bohrok]

Your reasoning for 1 is correct (the same reasoning applies to 5; that one has the inequality in the wrong direction).
2, 3, and 4 are correct too.

 

[ani9890]

thank you for your help,
so I just wanted to make sure, for my answers:

1. incorrect
2. correct
3. correct
4. incorrect
5. incorrect
6. correct

is this okay?

 

[Bohrok]

Looks good