Logarithmic divergences in the $k$-inflationary power spectra computed
  through the uniform approximation
  We investigate a calculation method for solving the Mukhanov-Sasaki equation
in slow-roll $k$-inflation based on the uniform approximation (UA) in
conjunction with an expansion scheme for slow-roll parameters with respect to
the number of $e$-folds about the so-called \textit{turning point}. Earlier
works on this method has so far gained some promising results derived from the
approximating expressions for the power spectra among others, up to second
order with respect to the Hubble and sound flow parameters, when compared to
other semi-analytical approaches (e.g., Green's function and WKB methods).
However, a closer inspection is suggestive that there is a problem when
higher-order parts of the power spectra are considered; residual logarithmic
divergences may come out that can render the prediction physically
inconsistent. Looking at this possibility, we map out up to what order with
respect to the mentioned parameters several physical quantities can be
calculated before hitting a logarithmically divergent result. It turns out that
the power spectra are limited up to second order, the tensor-to-scalar ratio up
to third order, and the spectral indices and running converge to all orders.
This indicates that the expansion scheme is incompatible with the working
equations derived from UA for the power spectra but compatible with that of the
spectral indices. For those quantities that involve logarithmically divergent
terms in the higher-order parts, existing results in the literature for the
convergent lower-order parts calculated in the equivalent fashion should be
viewed with some caution; they do not rest on solid mathematical ground.
