Series Expansion of Bessel Function of the First Kind

Theorem
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.

Then:

where $n$ is not a (strictly) negative integer.

Also see

 * Bessel Function of the First Kind of Negative Integer Order for when $n \in \set {-1, -2, -3, \ldots}$