Comparison test $\frac{n^2}{n^3+1}$

Question

I am working through the above problem and have not gotten too far:

for large n $$\frac{n^2}{n^3+1} \approx \frac{n^2}{n^3} = \frac{1}{n}$$

Now the next step in the solution given to me was:

$$\frac{n^2}{n^3+1} \ge \frac{n^2}{n^3+n^3} = \frac{1}{2n}$$

I understand the aim is to compare our unknown with a known series but I don't understand the reasoning behind this line. Where did the second $n^3$ come from and what happened to the $+1$?

Thank you

Answer 1

If $ n \ge 1 $, then $ n^3 \ge 1 $. Increasing the denominator decreases the value, so:

$$ \frac{n^2}{n^3 + 1} \ge \frac{n^2}{n^3 + n^3} $$