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We have applied the Lindstedt-Poincar\'e method to study the limit cycle ofthe van der Pol oscillator, obtaining the numerical coefficients of the seriesfor the period and for the amplitude to order $859$. Hermite-Pad\'eapproximants have been used to extract the location of the branch cut of theseries with unprecendented accuracy ($100$ digits). Both series have then beenresummed using an approach based on Pad\'e approximants, where the exactasymptotic behaviors of the period and the amplitude are taken into account.Our results improve drastically all previous results obtained on this subject.

High order analysis of the limit cycle of the van der Pol oscillator