What test is this am I missing something?

Question

$$\sum^{\infty}_{n = 0}{5n^{3} \over 2n^{3} +1}$$

$$ {5n^{3} \over 2n^{3} +1}<$$$${5n^{3} \over 2n^{3} } <$$ $$ {5 \over {2} }\ne 0$$

so the series is divergent but what test is this? am I missing something?

Answer 1

You are correct that the sum diverges. For large $n$, each term is about $\frac 52$. The logic of your steps is not clear at all. Some words would help. It appears you are using a theorem that says the limit of the terms must be zero for the sum to converge, but you don't say that.

Answer 2

Two things here, first I think there is a typo in your last limit sentence. The 3 should be a 2. Second, this is not a clean way of establishing that the answer is divergent. Leaving off the 1 is not very proper. Dividing every term by $n^3$ and then take the limit is a better way