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The recursive relation starts to appear by putting a Frobenius series into aFuchsian differential equation. There can be $d$-term of sequences in therecurrence relation of the Frobenius series where $d\geq 2$.  For equal and greater than a three-term case, various scholars have used aPoincar\'{e}--Perron (P-P) theorem in order to obtain the radius ofconvergence. We show that the P-P theorem only gives us some sort ofconditional-type convergence if it is applied to a Fuchsian equation. Thetheorem does not provide us absolute convergence. Moreover, we consider hereinthe absolute convergence test to the $d$-term recurrence relation withnon-constant coefficients of the Fuchsian equation.

Domain of the Frobenius series of the Fuchsian equation in relation to a Poincar\'{e}--Perron theorem