\[\frac{{{\pi ^2}}}{4} = 2 + \frac{1}{{1 + \frac{1}{2} + \frac{1}{{\frac{1}{2} + \frac{1}{3} + \frac{1}{{\frac{1}{3} + \frac{1}{4} + \frac{1}{{\frac{1}{4} + \frac{1}{5} + \frac{1}{{...}}}}}}}}}}\]