We consider the class of noisy multi-layered sigmoid recurrent neural networks with $w$ (unbounded) weights for classification of sequences of length $T$, where independent noise distributed according to $\mathcal{N}(0,σ^2)$ is added to the output of each neuron in the network. Our main result shows that the sample complexity of PAC learning this class can be bounded by $O (w\log(T/σ))$. For the non-noisy version of the same class (i.e., $σ=0$), we prove a lower bound of $Ω(wT)$ for the sample complexity. Our results indicate an exponential gap in the dependence of sample complexity on $T$ for noisy versus non-noisy networks. Moreover, given the mild logarithmic dependence of the upper bound on $1/σ$, this gap still holds even for numerically negligible values of $σ$.