We study singularly perturbed inhomogeneous linear systems which have a regular singularity at the origin. Under some extra assumption, this system has a unique formal solution which is a power series in the variable z and the perturbation parameter ε. Here we investigate its summability properties. In particular, we show that the series, regarded as a power series in ε with coefficients depending on z, is 1-summable under some eigenvalue condition. Moreover, we show that this condition in general is sharp.
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