Chapter 11.8, Problem 35E

The function $J_1$ defined by

$J_1\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n{x}^{2n+1}}{n!\left(n+1\right)!2^{2n+1}}}$

is called the Bessel function of order 1.

(a) Find its domain.

(b) Graph the first several partial sums on a common screen.

(c) If your CAS has built-in Bessel functions, graph $J_1$ on the same screen as the partial sums in part (b) and observe how the partial sums approximate $J_1$.