  Recently, cosmography emerged as a valuable tool to effectively describe the
vast amount of astrophysical observations without relying on a specific
cosmological model. Its model-independent nature ensures a faithful
representation of data, free from theoretical biases. Indeed, the commonly
assumed fiducial model, the $\Lambda$CDM, shows some shortcomings and tensions
between data at late and early times that need to be further investigated. In
this paper, we explore an extension of the standard cosmological model by
adopting the $f(z)$CDM approach, where $f(z)$ represents the cosmographic
series characterizing the evolution of recent universe driven by dark energy.
To construct $f(z)$, we take into account the Pad\'e series, since this
rational polynomial approximation offers a better convergence at high redshifts
than the standard Taylor series expansion. Several orders of such an
approximant have been proposed in previous works, here we want to answer the
questions: What is the impact of the cosmographic series choice on the
parameter constraints? Which series is the best for the analysis? So, we
analyse the most promising ones by identifying which order is preferred in
terms of stability and goodness of fit. Theoretical predictions of the
$f(z)$CDM model are obtained by the Boltzmann solver code and the posterior
distributions of the cosmological and cosmographic parameters are constrained
by a Monte Carlo Markov Chains analysis. We consider a joint data set of cosmic
microwave background temperature measurements from the Planck collaboration,
type Ia supernovae data from the latest Pantheon+ sample, baryonic acoustic
oscillations and cosmic chronometers data. In conclusions, we state which
series can be used when only late time data are used, while which orders has to
be considered in order to achieve the necessary stability when large redshifts
are considered.
