We extract the strong coupling constant α_s(m_τ^2) from the semileptonic τ decay width taking into account resummation effects from the running of the strong coupling constant. In the MS scheme the result reads α_s=0.375± 0.007 to third order and α_s=0.378± 0.007 to fourth order in the β-function, respectively, where we use the recently computed four-loop coefficient β_3. These values for the coupling constant have to be compared with the value α_s=0.354± 0.005 derived from a third order

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... s of τ decays. We determine the exact value of the convergence radius of the perturbation series by analyzing the singularity structure of the complex coupling constant plane.