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According to this answer, $$z_\infty:=\lim_n z_n=I:=\int_0^\infty F(s)G(s)\,ds,$$ where $$F(s):=\prod_{k=1}^\infty\frac{1}{\sqrt{1+2s/k^3}},\quad G(s):=\sum_{k=1}^\infty\frac k{k^3+2s}.$$ Mathematica can express $F$ and $G$ in terms of functions built-in in Mathematica (and these expressions should be rather straightforward to verify), and then the ...

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