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 $y(x)={\sum }_{n=0}^{\mathrm{\infty }}{d}_n{x}^n$ into a linear ordinary differential equation (ODE). There can be $d$-term of sequences in the recurrence relation of a power series where $d\ge 2$. We discuss the absolute convergence test to the $d$-term recurrence relation with non-constant coefficients of a linear ODE.