Mathematica incorrectly claims an infinite sum doesn't converge

Question

When I compare the output of the following commands I get contradictory results:

Sum[BesselJ[k, 1], {k, 0, Infinity}]
NSum[BesselJ[k, 1], {k, 0, Infinity}]

The former warns me that the sum does not converge and spits back the sum, while the latter outputs a number. I expect the sum to converge given the rapid decay of Bessel functions with order (see, for example, equation 10.19.1 of https://dlmf.nist.gov/10.19), so I think the former is giving me the wrong message. Why is it doing this and does this mean I have no hope of evaluating such infinite sums analytically in Mathematica?

Thanks in advance for any help.

Answer 1

Looks like Mathematica dosen't know how compute this series.

Workaround:

F = Sum[LaplaceTransform[BesselJ[k, x], x, s], {k, 0, Infinity}] // FullSimplify (*Where: x=1*)
(*(1 + s + Sqrt[1 + s^2])/(2 s Sqrt[1 + s^2])*)

InverseLaplaceTransform[F, s, x] /. x -> 1 // FullSimplify // Expand
(*1/2 + 1/2 BesselJ[0, 1] + 1/4 \[Pi] BesselJ[0, 1] StruveH[-1, 1] + 
1/4 \[Pi] BesselJ[1, 1] StruveH[0, 1]*)

% // N
(*1.34246*)

NSum[BesselJ[k, 1], {k, 0, Infinity}]
(*1.34246*)