MHB Determine whether the following SERIES converge or diverge

[shamieh]

can someone check my answers

Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.

4.$$\sum^{\infty}_{n=1} \frac{n^3}{3^n}$$

A: So for this one i used the ratio test and found that L = 1/3 which is < 1 therefore the series converges absolutely

5. $$\sum^{\infty}_{n = 1} \frac{n}{n^2 + 1}$$

A: diverges by limit comparison test

 

[Evgeny.Makarov]

4.$$\sum^{\infty}_{n=1} \frac{n^3}{3^n}$$

A: So for this one i used the ratio test and found that L = 1/3 which is < 1 therefore the series converges absolutely

Correct. (There is no need to add "absolutely" here since all terms are positive.)

5. $$\sum^{\infty}_{n = 1} \frac{n}{n^2 + 1}$$

A: diverges by limit comparison test

The problem statement asked to explain your answer. Mentioning the limit comparison test but not saying with what series you compare the given one is not a sufficient explanation. It would be nice to also say how you know that the second series converges or diverges.