The convergence test to the application in a multi-term recurrence relation of a linear ODE

The recursive relation starts to appear by putting a function $y(x)=\sum_{n=0}^{\infty }d_n x^n$ into a linear ordinary differential equation (ODE). There can be $d$-term of sequences in the recurrence relation of a power series where $d\geq 2$. We discuss the absolute convergence test to the $d$-term recurrence relation with non-constant coefficients of a linear ODE.