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Recently, cosmography emerged as a valuable tool to effectively describe thevast amount of astrophysical observations without relying on a specificcosmological model. Its model-independent nature ensures a faithfulrepresentation of data, free from theoretical biases. Indeed, the commonlyassumed fiducial model, the $\Lambda$CDM, shows some shortcomings and tensionsbetween data at late and early times that need to be further investigated. Inthis paper, we explore an extension of the standard cosmological model byadopting the $f(z)$CDM approach, where $f(z)$ represents the cosmographicseries characterizing the evolution of recent universe driven by dark energy.To construct $f(z)$, we take into account the Pad\'e series, since thisrational polynomial approximation offers a better convergence at high redshiftsthan the standard Taylor series expansion. Several orders of such anapproximant have been proposed in previous works, here we want to answer thequestions: What is the impact of the cosmographic series choice on theparameter constraints? Which series is the best for the analysis? So, weanalyse the most promising ones by identifying which order is preferred interms of stability and goodness of fit. Theoretical predictions of the$f(z)$CDM model are obtained by the Boltzmann solver code and the posteriordistributions of the cosmological and cosmographic parameters are constrainedby a Monte Carlo Markov Chains analysis. We consider a joint data set of cosmicmicrowave background temperature measurements from the Planck collaboration,type Ia supernovae data from the latest Pantheon+ sample, baryonic acousticoscillations and cosmic chronometers data. In conclusions, we state whichseries can be used when only late time data are used, while which orders has tobe considered in order to achieve the necessary stability when large redshiftsare considered.

Beyond $\Lambda$CDM with $f(z)$CDM: criticalities and solutions of Pad\'e Cosmography