How to compute $\prod\limits^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$ 
How to compute
  $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$

My Working : 
$$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1}
=  1-0 = 1$$
Is it correct

HINT:
If $$t_n=\frac{n^3-1}{n^3+1}=\frac{(n-1)(n^2+n+1)}{(n+1)(n^2-n+1)}$$
$$t_{n+1}=\frac{n\{(n+1)^2+n+1+1\}}{(n+2)\{(n+1)^2-(n+1)-1\}}=\frac{n\{(n+1)^2+n+1+1\}}{(n+2)(n^2+n+1)}$$
and $$t_{n-1}=\frac{(n-2)\{(n-1)^2+n-1+1\}}{n\{(n-1)^2-(n-1)-1\}}=\frac{(n-2)(n^2-n+1)}{n\{(n-1)^2-(n-1)-1\}}$$

Alternatively, let $\displaystyle u_n=\frac{n-1}{n+1},v_n=\frac{n^2+n+1}{n^2-n+1}$ so that $t_n=u_n\cdot v_n$
$$\implies\prod_{2\le n\le r}u_n=\frac{1\cdot2\cdot3\cdots(r-2)(r-1)}{3\cdot4\cdot5\cdots r(r+1)}\frac2{r(r+1)}$$
$$\implies\prod_{2\le n\le r}v_n=\frac{7\cdot13\cdot21\cdots(r^2-r+1)(r^2+r+1)}{3\cdot7\cdot13\cdots \{(r-1)^2-(r-1)+1\}(r^2-r+1)}=\frac{r^2+r+1}3$$
$$\implies\prod_{2\le n\le r} t_n=\left(\prod_{2\le n\le r}u_n\right)\left( \prod_{2\le n\le r}v_n\right)=\frac{2(r^2+r+1)}{3r(r+1)} $$
Setting $r\to\infty,$  $\displaystyle \frac{2(r^2+r+1)}{3r(r+1)}=\frac23\lim_{r\to\infty}\frac{1+\frac1r+\frac1{r^2}}{1+\frac1r}=?$
