<s>[INST] Continued fraction of $π$ using sums of cubes Recently I came across this identity:
$$\pi=3+\cfrac1{6+\cfrac{1^3+2^3}{6+\cfrac{1^3+2^3+3^3+4^3}{6+\cfrac{1^3+2^3+3^3+4^3+5^3+6^3}{6+\ddots}}}},$$ 
thus
$$\pi=3+\cfrac{1}{6+\cfrac{(1\cdot3)^2}{6+\cfrac{(2\cdot5)^2}{6+\cfrac{(3\cdot7)^2}{6+\ddots}}}}.$$
We also know that
$$\pi=3+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(n+1)(2n+1)},$$
Converting the sum to a continued fraction we get:
$$\pi=3+\cfrac{1}{6+\cfrac{(1\cdot3\cdot2)^2}{6\cdot2^2+\cfrac{(2\cdot5\cdot3)^2}{6\cdot3^2+\cfrac{(3\cdot7\cdot4)^2}{6\cdot4^2+\ddots}}}}.$$
How are the two continued fractions equal? And how is the above continued fraction using sums of cubes derived?
 [/INST] The (first) identity is wrong; the continued fraction with "sums of cubes" doesn't converge. This can be shown using the known criterion: for a sequence $\{a_n\}$ of positive real numbers, the continued fraction $$a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\ldots}}}$$ converges if and only if the series $\sum_{n=0}^{\infty}a_n$ diverges. In our case, for $n>0$, we have $a_n=6c_n$ where $c_1=1$ and $c_{n+1}=1/\big(n^2(2n+1)^2 c_n\big)$, and one obtains $\color{blue}{a_n=\mathcal{O}(1/n^2)}$ using, say, the $\Gamma$-function: $$c_n=d_n\left(\frac{\Gamma\left(\frac{n}{2}\right)\Gamma\left(\frac{n}{2}+\frac14\right)}{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{n}{2}+\frac34\right)}\right)^2\quad\implies\quad d_{n+1}=\frac{1}{64d_n},$$ giving $d_n=\mathcal{O}(1)$ and $c_n=\mathcal{O}(1/n^2)$ since $\Gamma(x+a)/\big(x^a\Gamma(x)\big)\underset{x\to\infty}{\longrightarrow}1$.
For curiosity, here are computed approximate values of the lower/upper limits, respectively:
$$\mathtt{3.14221404702232210406367353362166370131484883936217-}\\\mathtt{3.15126273205858662275081482878228893534757749143403-}$$
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