<s>[INST] If $\frac {a_{n+1}}{a_n} \ge 1 -\frac {1}{n} -\frac {1}{n^2}$ then $\sum\limits_na_n$ diverges 
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that, for every $n\ge1$,
  $$\frac {a_{n+1}}{a_n} \ge 1 -\frac {1}{n}  -\frac {1}{n^2} \tag 2$$ Prove that $x_n=a_1 + a_2 + .. + a_n$ diverges.

It is clear that $x_n$ is increasing, so it has to have a limit. I tried to prove the limit is $+\infty$ but without success. No divergence criteria from series seems to work here.
UPDATE
Attempt: 
Suppose a stronger inequality holds, namely that, for every $n\ge1$, $$\frac{a_{n+1}}{a_n}\geqslant1-\frac1n \tag 1$$ Then:
$$\frac {a_3}{a_2} \ge \frac 1 2\qquad
\frac {a_4}{a_3} \ge \frac 2 3\qquad
\ldots\qquad
\frac {a_{n-1}}{a_{n-2}} \ge \frac {n-3}{n-2}\qquad
\frac {a_n}{a_{n-1}} \ge \frac {n-2}{n-1}$$
Multiplying all the above yields
$$\frac {a_n}{a_2} \ge \frac 1 {n-1}$$
The last inequality proves the divergence.
 [/INST] You may notice that
$$ \frac{a_{n+1}}{a_n} \geq \left(1-\frac{1}{n}\right)\left(1+\frac{1}{n^2-n-1}\right)^{-1} \tag{1}$$
hence:
$$ \frac{a_{N+1}}{a_2}\geq \frac{1}{N}\prod_{n=2}^{N}\left(1+\frac{1}{n^2-n-1}\right)^{-1} \tag{2}$$
but the infinite product $\prod_{n\geq 2}\left(1+\frac{1}{n^2-n-1}\right)^{-1}$ is convergent to a positive number ($-\frac{1}{\pi}\cos\frac{\sqrt{5}}{2}\approx 0.296675134743591$). It follows that $a_{N+1}\geq \frac{C}{N}$ so $\sum_{n\geq 2}a_n$ is divergent.
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