Some observations on a Kapteyn series

We study the Kapteyn series $% %TCIMACRO{\dsum \limits_{n=1}^{\infty}}% %BeginExpansion {\displaystyle\sum\limits_{n=1}^{\infty}} %EndExpansion t^{n}$\mathrm{J}$_{n}(nz) $\mathrm{.}Wefindaseriesrepresentationinpowersof$z$ and analyze its radius of convergence.