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We say that the pair $(E,F)$, for Fréchet spaces $E$ and $F$, is called a tame pair if there exists an increasing function $S:\mathbb{N}\to \mathbb{N}$ such that for any linear operator $T:E\to F$, we have $\pi _T(k)\leq S(k)$ when $k$ is large enough, where $\pi _T(k)$ is the characteristic of continuity map of $T$. We show that, for K\"othe spaces, the pair $(\lambda(A),\lambda(B))$ is tame if and only if the family of quasidiagonal operators from $\lambda(A)$ to $\lambda(B)$ satisfies the tameness criteria. We give tameness characterization for power series spaces of finite and infinite types. It turns out that if $\alpha$ or $\beta$ is stable, then the tameness of the pair $(\Lambda_\infty(\alpha),\Lambda_\infty(\beta))$ is equivalent to boundedness. We also characterize the tameness of mixed type power series spaces.