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The following result is proved: Let ${\Lambda }_R^p\left(\alpha \right)$ denote a power series space of infinite or of finite type, and equip ${\Lambda }_R^p\left(\alpha \right)$ with its canonical fundamental system of norms, R ∈ 0,∞, 1 ≤ p < ∞. Then a tamely exact sequence (⁎) $0\to {\Lambda }_R^p\left(\alpha \right)\to {\Lambda }_R^p\left(\alpha \right)\to {\Lambda }_R^p{\left(\alpha \right)}^{\mathbb{N}}\to 0$ exists iff α is strongly stable, i.e. $li{m}_n{\alpha }_{2n}/{\alpha }_n=1$, and a linear-tamely exact sequence (*) exists iff α is uniformly stable, i.e. there is A such that $limsu{p}_n{\alpha }_{Kn}/{\alpha }_n\le A<\infty$ for all K. This result extends a theorem of Vogt and Wagner which states that a topologically exact sequence (*) exists iff α is stable, i.e. $su{p}_n{\alpha }_{2n}/{\alpha }_n<\infty$.