TITLE: How to prove divergent sequences?
QUESTION [1 upvotes]: For this question, I know that the sequence diverges to infinity, but I'm not sure if I am doing it right. Here is what I have so far. Can anyone please help me out?
Determine whether the following sequence is convergent or divergent
$a_n = \{8n^3 + n^2 -2\}$
$\lim_{n \to \infty} a_n = \infty$
Wts for any $M>0$, there exists some $N>0$, st if $n>N$, then $a_n>M$.
$n^3 > N^3 > M$
$n^3(8 + \frac{1}{n} - \frac{2}{n^3}) > N^3(8 + \frac{1}{N} - \frac{2}{N^3}) > M$
$n^3(8 + \frac{1}{n} - \frac{2}{n^3}) > N > (\frac{M}{8+\frac{1}{N} -\frac{2}{N^3}})^\frac{1}{3}$

REPLY [0 votes]: A different approach: Observe that your sequence is strictly increasing and integer-valued. Thus $a_{n}\geq a_{n-1}+1$, and so $a_n\geq a_0+n$. Now you should be able to easily prove $a_n$ is unbounded and hence divergent.
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