Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? 
In process of solving this problem, I faced to the problem of proving that
$A::$:  $\dfrac{2k+1}{(k(k+1))^2}-\dfrac{4}{3^{k+1}}\geq 0$ for every positive odd integer $k>1$, 
which will complete the proof that ${\{a_n}\}$ would be monotonically decreasing. 
Even induction method makes it more complicated. Is statement $A$ holds?
Thank you.  

Note first that $$\dfrac{2k+1}{(k(k+1))^2}\geqslant \dfrac{2k}{k^2(k+1)^2}= \dfrac{2}{k(k+1)^2}\geqslant \dfrac{2}{(k+1)^3}$$
If we prove that
$$\dfrac{1}{n^3}\geqslant \dfrac{2}{3^n} \tag{1}$$
which is equivalent to
$$\dfrac{3^n}{n^3} \geqslant  2 \tag{2}$$
then we can conclude that
$$\dfrac{2k+1}{(k(k+1))^2}-\dfrac{4}{3^{k+1}}\geqslant 0.$$
The inequality $(2)$ holds for sufficiently large $n,\;\;(n\geqslant{6})$ and can be proved by induction.
