As far as series approximation and function values go, the Bessel function is a nightmare, but it does mean something physical, especially in relation to quantum mechanics. So yes, calculate!
In some sense it is like the cosine function, where series approximation is already made difficult by the fact that the range never exceeds
$\pm 1$, while the domain goes between
$\pm \infty$. At the outer limits all polynomials blow up, so it's hopeless to translate to another function value via series expansion around some starting point.
Considering any function as a manifold, series expansion is a local technique, to combine with overall global structure. You can get a pretty good approximation of sine / cosine if you expand a few orders of magnitude around say, every minima and maxima at intervals of
$\pi$.
This technique also applies to the Bessel function, another example where
$$\lim_{x\rightarrow\infty}\frac{y(x)}{x} \longrightarrow 0 . $$
Mathematica greatly simplifies the finding of zeroes,
ApproxBessel[nCut_] := Normal@Series[BesselJ[0, x], {x, BesselJZero[0, #], nCut}] & /@ Range[5]
FList = N[ApproxBessel[#]] & /@ Range[10];
gList = MapThread[ Plot[#1, {x, 0, 20}, PlotRange -> {-1.1, 1.1}, PlotStyle -> #2,
Axes -> False] &, { FList, Blend[{Green, Blue}, #/11] & /@ Range[10] }];
Stacked = Show[gList[[1 ;; #]]] & /@ Range[10];
ListAnimate[ Join[Stacked, Reverse@Stacked]]
We can also approach critical values from either the left or right,
In[42]:= x /. FindRoot[#, {x, 4}] & /@ D[FList[[-1, {1, 2}]], x]
Out[42]= {3.83171, 3.83168}
Then compare with the first entry of the last table on mathworld: Bessel Function Zeroes . Matching of left / right values essentially provides a convergence criteria that allows you to say if a digit in decimal is correct. Up to order ten, we have four digits correct on the first minima. Getting a few of these values, we could make another patch function by expanding in quadratic and higher.
