I know from my intuition that the sequence
$$x_n=\left(1-\cfrac{1}{3}\right)^2 \left(1-\cfrac{1}{6}\right)^2 \left(1-\cfrac{1}{10}\right)^2\cdots \cdots\left(1-\cfrac{1}{\cfrac{n\left(n+1\right)}{2}}\right)^2,\quad n\geq2$$
is convergent. But i don't know how to prove it.I almost try to apply every theorem I know (for eg ratio test ,monotone convergence theorem,...). Help me to prove this.
Proof or idea is needed.Where does the sequence converge to?
$$ \begin{align} \prod_{k=2}^\infty\left(1-\frac2{k^2+k}\right) &=\lim_{n\to\infty}\prod_{k=2}^n\frac{\color{#C00}{(k-1)}\color{#090}{(k+2)}}{\color{#00F}{k}\color{#C90}{(k+1)}}\\ &=\lim_{n\to\infty}\frac{\color{#C00}{(n-1)!}\,\color{#090}{(n+2)!/3!}}{\color{#00F}{n!/1!}\,\color{#C90}{(n+1)!/2!}}\\[3pt] &=\frac13\lim_{n\to\infty}\frac{n+2}{n}\\[3pt] &=\frac13 \end{align} $$ Your product is just the square of this one.
$$\prod_{k=2}^n\left(1-\frac{2}{k(k+1)}\right)=\prod_{k=2}^n\frac{(k+2)(k-1)}{k(k+1)}=$$ $$\frac{4\cdot1}{2\cdot3}\cdot\frac{5\cdot2}{3\cdot4}\cdot\frac{6\cdot3}{4\cdot5}\cdot...\cdot\frac{n(n-3)}{(n-2)(n-1)}\cdot\frac{(n+1)(n-2)}{(n-1)n}\cdot\frac{(n+2)(n-1)}{n(n+1)}=\frac{n+2}{3n}$$
HINT:
$$1-\dfrac2{n(n+1)}=\dfrac{n^2+n-2}{n(n+1)}=\dfrac{(n+2)(n-1)}{n(n+1)}=\dfrac{\dfrac{n-1}n}{\dfrac{n+1}{n+2}}=\dfrac{f(n-1)}{f(n+1)}$$
where $f(m)=\dfrac m{m+1}$
prove by induction that for your sum is hold $$\frac{(n+2)^2}{9n^2}$$
