  We propose a generalized formula for calculating the dipole polarizability of
spherical multilayer nanoshells (MNSs) within the long-wavelength approximation
(LWA). Given a MNS with a finite number of concentric layers, radii, and
dielectric properties, embedded in a dielectric medium, in the presence of a
uniform electric field, we show that its frequency-dependent and complex dipole
polarizability can be expressed in terms of the dipole polarizability of the
preceding MNS. This approach is different from previous more involved methods
where the LWA polarizability of a MNS is usually derived from scattering
coefficients. Using both finite-element method- and Mie theory-based
simulations, we show that our proposed formula reproduces the usual LWA
results, when it is used to predict absorption spectra, by comparing the
results to simulated spectra obtained from MNSs with $n$ number of layers up to
$n = 6$ layers. An iterative algorithm for calculating the dipole
polarizability of a MNS based on the generalized formula is presented. A
Fr\"{o}hlich function whose zeroes correspond to the dipolar localized surface
plasmon resonances (LSPRs) supported by the MNS is proposed. We identify a
pairing behaviour by some LSPRs in the Fr\"{o}hlich function that might also be
useful for mode characterization.
