The exact analytical result for any continued fraction coefficients

In this chapter, the analysis is aimed at deriving the exact general analytical expression for all the expansion coefficients a_n in the CF. To this end we shall use the theorem of uniqueness of two power series expansions for the same function. The expressions for the coefficients a₁ and a₂ are derived by starting from the defining expression. The third coefficient a₃ is calculated from the third-order CF. The fourth coefficient a₄ is calculated using the fourth-order CF_{. The calculation of the fifth coefficient a5 is performed by means of the fifth-order CF. In order to obtain the sixth coefficient a6 the sixth-order CF is employed. A derivation of the expression for the seventh coefficient a7 necessitates the seventh-order CF.}