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The [[radius of convergence]] of this series is <math>e^{-1}</math> (giving the [[principal branch]] of the Lambert function).
A series that converges for <math>|\ln(z)-1|<{4+\pi^2}</math> (approximately <math>2.58\ldots \cdot 10^{-6} < z < 2.869\ldots \cdot 10^6</math>) can also be derived by series inversion. The function <math>f(z) = W(e^z) - 1</math> satisfies the equation
:<math>1 + f(z) + \ln (1 + f(z)) = z.</math>
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