  In recent work on the area of approximation methods for the solution of
nonlinear differential equations, it has been suggested that the so-called
generalized Taylor series approach is equivalent to the homotopy analysis
method. In the present paper, we demonstrate that such a view is only valid in
very special cases, and in general the homotopy analysis method is far more
robust. In particular, the equivalence is only valid when the solution is
represented as a power series in the independent variable. As has been shown
many times, alternative basis functions can greatly improve the error
properties of homotopy solutions, and when the base functions are not
polynomials or power functions, we no longer have that the generalized Taylor
series approach is equivalent to the homotopy analysis method. We demonstrate
this by consideration of an example where the generalizes Taylor series must
always have a finite radius of convergence (and hence limited applicability),
while the homotopy solution is valid over the entire infinite domain. We then
give a second example for which the exact solution is not analytic, and hence
it will not agree with the generalized Taylor series over the domain. We
conclude that the generalized Taylor series approach is not equivalent to the
homotopy analysis method. In particular, the generalized Taylor series can at
best recover local information about solutions, whereas the homotopy analysis
method can recover the solutions globally if appropriate base functions are
selected. Such results have important implications for how series are
calculated when approximating solutions to nonlinear differential equations.
