Which mathematical observations about the transcendence of the limit of the studied series can we make?

Question

Aufgabe: Generalized formulas for all rational numbers B with B>1.

Problem/Ansatz:

Daniel Bertrand discussed the transcendence of Theta functions especially Theta-3. However, my article proves also that four infinite series which have the same limit are equivalent and one of the equivalent series is a product of two terms. Which mathematical observations about the transcendence of the limit of the studied series can we make? I Generalized my équation to all rational numbers B with B>1.

Here is again the link to my article :

https://www.researchgate.net/publication/364651911_A_useful_new_equation_of_four_infinite_series_and_sums_by_using_a_new_demonstrated_recurrence_relation