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Certain numerical methods for initial value problems have as stability function the n th partial sum of the exponential function. We study the stability region, i.e., the set in the complex plane over which the n th partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semi-disk. We quantify this by providing disks that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disk centered at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 ≤ n ≤ 20. Bounds on such radii are proved for n ≥ 2; these bounds are shown to be optimal in the limit n → +∞. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to π, as n → ∞. Finally, we prove that a semi-disk in the left half-plane with vertical boundary being the imaginary axis and centered at the origin is included in the stability region if and only if n ≡ 0 mod 4 or n ≡ 3 mod 4. The maximal radii of such semi-disks are exactly determined for 1 ≤ n ≤ 20.

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