The question provides the series definition for the Bessel function of the first kind of order :

The question provides the series definition for the Bessel function of the first kind of order $\nu$:

$$J_{\nu}(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{2n-\nu} n! \Gamma(n-\nu+1)} x^{2n-\nu}$$

This is the definition as given in the problem statement.

The given Bessel equation is:

$$x^2 y'' + xy' + (x^2 - \nu^2)y = 0$$

We assume a series solution of the form