WKB to all orders and accuracy of the semiclassical quantization

We perform a systematic WKB expansion to all orders for a one–dimensional system with potential $V(x)=U_0\mathrm{/}{\cos }^2(\alpha x)$. We are able to sum the series to the exact energy spectrum. Then we show that any finite order WKB approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.