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Q:

showing that $\sum_{n=0}^{\infty} \frac{n^3}{(1+2n)^3}$ converges

Let $$S=\sum_{n=0}^{\infty} \frac{n^3}{(1+2n)^3}.$$
Show that $S$ converges.
I know that my method here is to find a lower bound and an upper bound to the series, but i am unable to do so, I don’t know how to proceed with this problem, and what i did before was to use $\sum_{n=0}^{\infty} \frac{n}{(1+2n)^3}$, but this time i am not able to bound it.
If we want to solve it, we have to prove that the sequence is monotone decreasing, then it is decreasing, hence it is bounded above, then it has to be less than the limit, which is $\infty$. How to do this?
EDIT: i solved the problem, i will post my solution and i would like to have more feedback about my solution. Here is the solution:
$S$ is convergent iff there exists an $M\in\mathbb{R}$ such that $\forall n\in\mathbb{N}$, $\frac{n^3}{(1+2n)^3}

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