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A series solution about an ordinary point of a differential equation is always a Taylor series having a nonvanishing radius of convergence. A series solution about a singular point does not have this form (except in rare cases). Instead, it may be either a convergent series not in Taylor series form (such as a Frobenius series) or it may be a divergent series.
Source: wiktionary
The usual procedure for deriving finite-difference equations consists of approximating the derivatives in the differential equation via a truncated Taylor series.
Source: wiktionary
This function has its only singularity at x = 0, implying that the radius of convergence for the Taylor series around x = 1 is only unity.
Source: wiktionary