  The Frobenius method can be used to compute solutions of ordinary linear
differential equations by generalized power series. Each series converges in a
circle which at least extends to the nearest singular point; hence
exponentially fast inside the circle. This makes this method well suited for
very-high-precision solutions of such equations. It is useful for this purpose
to have prior knowledge of the behaviour of the series. We show that the
magnitude of its coefficients can be apriori predicted to surprisingly high
accuracy, employing a Legendre transformation of the WKB approximated solutions
of the equation.
