I resolved this limit and got the solution $ frac {3} {4} $. I tried to check WolframAlpha, but when generating the presentation the expression shows $ lim {n to n} $ instead of $ lim {n to infty} $ and tells me that the limit diverges. So I am not sure if it diverges due to misunderstanding of the problem or if the limit is really divergent.

$$ lim_ {n to infty} left ( frac {n ^ 3} {1 cdot 3 + 3 cdot5 + cdots + (2n-1) (2n + 1)} right) $$

I applied the Stolz-Cesaro theorem and eventually (after the initial steps) I obtained

$$ lim_ {n to infty} frac {(n + 1) ^ 3-n ^ 3} {1 cdot 3 + 3 cdot5 + cdots + (2n-1) (2n + 1) – (1 cdot 3 + 3 cdot5 + cdots + (2n-1) (2n + 1) + (2n + 1) (2n + 3))} = lim_ {n to infty} frac {3n ^ 2 + 3n +1} {(2n + 1) (2n + 3)} = frac {3} {4} $$

Is the result correct?

Thanks in advance

P.S.

Should I eliminate these types of questions if the answer is a simple yes, since they do not provide much information and may not be very useful for anyone but me?