Use Fourier Bessel series and take n = 0 and R = 1 so that we get the series:

f(x) = a₁J₀ (_0,1x) + a₂J₀(_0,2x) + a₃J₀(_0,3x) + ... (1)

with the zeros _{0,1 0,2} from your CAS (Ref Bessel Function Tables)

Write a program for calculating partial sums of (1). Find out for what f(x) your CAS can evaluate the integrals. Take two such f(x) and comment empirically on the speed of convergence by observing the decrease of the coefficients.