Let \((X_n)_{n\in\mathbb N}\) be i.i.d.real valued random variables and assume that \((n^{1/\alpha}L(n))^{-1}\sum_{k=1}^nX_k\) with distribution function (df) \(F_n\) converges in distribution to a symmetric \(\alpha\)-stable random variable with df \(F\) for some \(0<\alpha\leq2\) and a non-trivial slowly varying function \(L\). The authors show that \(\limsup_{n\to\infty}\gamma_n/|1-L(n)/L(2n)|>0\) for either \(\gamma_n=\|F_n-F\|_\infty\) or \(\gamma_n=\|F_{2n}-F\|_\infty\). The same holds for corresponding density functions instead of df provided the law of \(X_1\) is absolutely continuous. From this the authors conclude that the rate is at best logarithmic, since for a slowly varying function \(\limsup_{n\to\infty}(\log n)^{1+\varepsilon}/|1-L(n)/L(2n)|>0\) if \(L(x)\to0\) or \(L(x)\to\infty\).