@dharr 

I'm attaching two spreadsheets (I hope I've done that correctly).  I have a new/obscure? representation for the Bessel function I_v(r) that leads to fairly easy to derive derivatives with respect to the order, for allow nonnegative integers v (as well as J_v(r) but that's irrelevant for now). 

In DI1, if compare my expansion to Maple's BesselI function for various parameter values of v and z.  I show that my series converges to the Maple values.  I take Maple's first derivatives of my series and compare it to my expression as a check. 
For order v=0, there is a well-klnown closed form for the first derivative and I check that I can correctly match it. 

DI2 looks at second derivatives and compares numerical estimates to my expression.  I've gone as high as 6th order comparing numerical derivatives to my formula and (if we increase Digits), everything looked fine.  But I can't seem to write a procedure that evaluates my infinite sum for an arbitrary higher order derivative.  I tried computing the values of the function phi for the first n terms given v and z and storing it in a Vector.  I then use a loop to compute the coefficients in the infinite Sum but Maple returns an error when I try to access the terms in the Vector within the loop (even if I surround it in quotes)

Download DI1.mw

Download DI2.mw

@Carl Love 

A paper by Li and Qin in Symmetry (2023) provides a recursive formula for calculating the derivatives of Bessel functions with respect to order.  I have a different recursive formula and I'd like to compare their properties.  A colleague has been able to program up both formulae in Mathematica but I'm not having the same success in Maple.....

@dharr 

I'm hoping for a procedure where I provide the value of the arguments (n,v,z) and have Maple return a numerical value for the infinite sum.  

@vv Divergent_series.mw

It turns out that there is a problem using the Borel transform--it has a singularity on the real positive axis--so that's definitely NOT what Maple is using.  Thanks for the info about the Levin transform--I'll try to learn more about how it works.

Mea culpa.  Thanks for catching that.

Thank you acer.  That's very informative.

Thanks for your help.  I thought it was OK to try symbolic integration and then numerical integration if that failed...

From Maple's help command (evalf int or Numerical Integration):

The most common command for numerical integration is evalf(Int(f, x=a..b)) where the integration command is expressed in inert form to avoid first invoking the symbolic integration routines. It is also possible to invoke evalf on an unevaluated integral returned by the symbolic int command, as in evalf(int(f, x=a..b)), if it happens that symbolic int fails (returns an unevaluated integral).

Based on previous experience, the statement that I highlighted in bold is usually correct!  But I guess the conditions under which "It is possible"  are more complicated than just symbolic int fails....

Thank you.  Your trick for avoiding evalhf seems unintuitive, but I'm impressed.  It works!!