Mr. IntegralPower-Series Solutions — Cheat sheet
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Core rule
Shift all sums to the same power before equating coefficients; use initial data to start the recurrence.
Watch for
A truncated series is an approximation and needs an error or convergence argument for its intended interval. Singular points may require a Frobenius series rather than ordinary nonnegative integer powers; that is a further topic beyond this introductory method.