Solution verification of a series with integer part

Question

I want to study $\sum_{n=1}^{\infty}(-1)^n \bigg\lfloor{\frac{n+2}{n^2}}\bigg\rfloor$. I have thought to apply the leibniz criterion but first of all I have to remark that:

1. $a_n=\lfloor{\frac{n+2}{n^2}}\rfloor\to 0$\
2. $a_n>0$
3. $a_n$ decreasing

For the last point I have considered that: $$\bigg\lfloor{\frac{n+2}{n^2}}\bigg\rfloor=\bigg\lfloor{\frac{1}{n}+\frac{2}{n^2}}\bigg\rfloor\geq \bigg\lfloor{\frac{1}{n+1}+\frac{2}{(n+1)^2}}\bigg\rfloor\,\, \forall n\geq 1$$ So this implies the decrease of $a_n$ and then the series converges.

Is my attempt correct?

Answer 1

Since $(-1)^n\left\lfloor\frac{n+2}{n^2}\right\rfloor=0$ when $n>2$, your series converge.

But, yes, your approach is correct.