I've been trying to solve this series of fractions and find the closed form of this sequence. The problem is that new ramifications of fractions are growing exponentially in each term and I have to add fractions with different denominator all the time, which makes more difficult the sequence as it continues because there are more and more and more ramifications...
$$\frac{1}{2},\frac{1+\frac{3}{4}}{2+\frac{5}{6}},\frac{1+\frac{3+\frac{7}{8}}{4+\frac{9}{10}}}{2+\frac{5+\frac{11}{12}}{6+\frac{13}{14}}}, \frac{1+\frac{3+\frac{7+\frac{15}{16}}{8+\frac{17}{18}}}{4+\frac{9+\frac{19}{20}}{10+\frac{21}{22}}}}{2+\frac{5+\frac{11+\frac{23}{24}}{12+\frac{25}{26}}}{6+\frac{13+\frac{27}{28}}{14+\frac{29}{30}}}},...,¿a_n?$$
Thank you!!