  The physical significance of tidal deformation in astronomical systems has
long been known. The recently discovered universal I-Love-Q relations, which
connect moment of inertia, quadrupole tidal Love number, and spin-induced
quadrupole moment of compact stars, also underscore the special role of tidal
deformation in gravitational wave astronomy. Motivated by the observation that
such relations also prevail in Newtonian stars and crucially depend on the
stiffness of a star, we consider the tidal Love numbers of Newtonian polytropic
stars whose stiffness is characterised by a polytropic index $n$. We first
perturbatively solve the Lane-Emden equation governing the profile of
polytropic stars through the application of the scaled delta expansion method
and then formulate perturbation series for the multipolar tidal Love number
about the two exactly solvable cases with $n=0$ and $n=1$, respectively. Making
use of these two series to form a two-point Pad\'e approximant, we find an
approximate expression of the quadrupole tidal Love number, whose error is less
than $2.5 \times 10^{-5}$ per cent (0.39 per cent) for $n\in[0,1]$
($n\in[0,3]$). Similarly, we also determine the mass moments for polytropic
stars accurately. Based on these findings, we are able to show that the
I-Love-Q relations are in general stationary about the incompressible limit
irrespective of the equation of state (EOS) of a star. Moreover, for the
I-Love-Q relations, there is a secondary stationary point near $n \approx
0.4444$, thus showing the insensitivity to $n$ for $n\in[0,1]$. Our
investigation clearly tracks the universality of the I-Love-Q relations from
their validity for stiff stars such as neutron stars to their breakdown for
soft stars.
