Infinite Fractions - 8

$\frac { 2+\frac { 2+\frac { 2+\cdots }{ { 2 }^{ 2 }+\cdots } }{ { 2 }^{ 2 }+\frac { 2+\cdots }{ { 2 }^{ 2 }+\cdots } } }{ { 2 }^{ 2 }+\frac { 2+\frac { 2+\cdots }{ { 2 }^{ 2 }+\cdots } }{ { 2 }^{ 2 }+\frac { 2+\cdots }{ { 2 }^{ 2 }+\cdots } } } =x$

Given that $x$ is positive, find $x$ .

Note : Round answer to $3$ decimal places.