text: A large part of Frobenius' 1873 publication was devoted to proofs of convergence of all the series involved in the solutions, as well as establishing the radii of convergence of these series.
  Explanation of Frobenius Method: first, linearly independent solution
  The method of Frobenius is to seek a power series solution of the form
  Differentiating:
  Substituting the above differentiation into our original ODE:
  The expression
  is known as the indicial polynomial, which is quadratic in r. The general definition of the indicial polynomial is the coefficient of the lowest power of z in the infinite series. In this case it happens to be that this is the rth coefficient but, it is possible for the lowest possible exponent to be r − 2, r − 1 or, something else depending on the given differential equation. This detail is important to keep in mind. In the process of synchronizing all the series of the differential equation to start at the same index value (which in the above expression is k = 1), one can end up with complicated expressions.  However, in solving for the indicial roots attention is focused only on the coefficient of the lowest power of z.
  Using this, the general expression of the coefficient of  is
  These coefficients must be zero, since they should be solutions of the differential equation, so
  The series solution with  above,