  A simple analytic tool namely the general series expansion method is proposed
to find the solutions for nonlinear differential equations. By choosing a set
of suitable basis functions $\{e_n(t,t_0)\}_{n=0}^{+\infty}$ such that the
solution to the equation can be expressed by
$u(t)=\sum_{n=0}^{+\infty}c_ne_n(t,t_0)$. In general, $t_0$ can control and
adjust the convergence region of the series solution such that our method has
the same effect as the homotopy analysis method proposed by Liao, but our
method is more simple and clear. As a result, we show that the secret parameter
$h$ in the homotopy analysis methods can be explained by using our parameter
$t_0$. Therefore, our method reveals a key secret in the homotopy analysis
method. For the purpose of comparison with the homotopy analysis method, a
typical example is studied in detail.
