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