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.