Let $S$ be Schwartz space defined in the following way: it consists of all such functions $f\in C^\infty(\mathbb R^n)$ that for any arbitraty multiindices $\alpha, \beta$ $$\exists M_{\alpha,\beta}: \,|x^\beta\partial^\alpha f(x)|<M_{\alpha,\beta} \,\forall x\in\mathbb R^n$$
We defined convergence in S in the following way:
$f_n \to f$ in $S$ iff $\|f_n-f\|_k=\max\limits_{|\alpha|\leq k}\sup\limits_{x \in \mathbb R^n} |(1+x)^k\partial^\alpha (f_n(x)-f(x))|\to0$ as $n \to \infty$ for any $k > 0$.
I would like to know if there are any sufficient conditions that guarantee convergence in S and do not require computation of all possible derivatives.
Are there such?