Question about some zeta function or Apery's constant

Question

I had found this question in MSE, which has one of its equation $$\sum_{i=1}^{\infty}\frac{(3\sqrt{3})^3x^3}{(6i+1)^3\pi^3}\cong\frac{2x^3}{5!}$$ Can anyone give a detailed explaination of this approximation?

Answer 1

$$S_n=\sum_{i=1}^{n}\frac{1}{(6i+1)^3}=\frac{1}{432} \left(\psi ^{(2)}\left(n+\frac{7}{6}\right)-\psi ^{(2)}\left(\frac{7}{6}\right)\right)$$ $$S_n=\frac{1}{432} \psi ^{(2)}\left(n+\frac{7}{6}\right)+\frac{91 }{216}\zeta (3)+\frac{\pi ^3}{36 \sqrt{3}}-1$$ When $n$ is large $$ \psi ^{(2)}\left(n+\frac{7}{6}\right)=-\frac 1{n^2}+\frac{4}{3 n^3}+O\left(\frac{1}{n^4}\right)$$ So $$S_\infty=\frac{91 }{216}\zeta (3)+\frac{\pi ^3}{36 \sqrt{3}}-1$$

$$\frac{(3\sqrt{3})^3}{\pi^3}S_\infty=\frac{3 \sqrt{3} \left(91 \zeta (3)-216+2 \sqrt{3} \pi ^3\right)}{8 \pi ^3}=0.016676\cdots$$ while $\frac 2{5!}=\frac 1{60}=0.016666\cdots$